In Our Time - Symmetry

Episode Date: April 19, 2007

Melvyn Bragg and guests discuss symmetry. Found in Nature - from snowflakes to butterflies - and in art in the music of Bach and the poems of Pushkin, symmetry is both aesthetically pleasing and an essential tool to understanding our physical world. The Greek philosopher Aristotle described symmetry as one of the greatest forms of beauty to be found in the mathematical sciences, while the French poet Paul Valery went further, declaring; “The universe is built on a plan, the profound symmetry of which is somehow present in the inner structure of our intellect”.The story of symmetry tracks an extraordinary shift from its role as an aesthetic model - found in the tiles in the Alhambra and Bach's compositions - to becoming a key tool to understanding how the physical world works. It provides a major breakthrough in mathematics with the development of group theory in the 19th century. And it is the unexpected breakdown of symmetry at sub-atomic level that is so tantalising for contemporary quantum physicists.So why is symmetry so prevalent and appealing in both art and nature? How does symmetry enable us to grapple with monstrous numbers? And how might symmetry contribute to the elusive Theory of Everything?With Fay Dowker, Reader in Theoretical Physics at Imperial College, London; Marcus du Sautoy, Professor of Mathematics at the University of Oxford; Ian Stewart, Professor of Mathematics at the University of Warwick.

This BBC podcast is supported by ads outside the UK. Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.uk forward slash radio 4. I hope you enjoy the program. Hello, today we'll be discussing symmetry, from the most perfect forms in nature like the snowflake and the butterfly, to our perceptions of beauty in the human face

and the deep symmetry in most of the laws that govern our physical world. The Greek philosopher Aristotle described symmetry as one of the greatest forms of beauty to be found in the mathematical sciences, while the French poet, Paul Valéry, went further declaring the universe is built on a plan, the profound symmetry of which is somehow present in the inner structure of our intellect.

The story of symmetry tracks an extraordinary shift from its role as an aesthetic model found in the tiles in the Halambra and Bach's compositions to becoming a key tool to understanding how the physical world works. It provides a major problem. breakthrough in mathematics with the development of group theory in the early 19th century, and it's the unexpected breakdown of symmetry at subatomic level that's so tantalizing for

contemporary quantum physics. So why is symmetry so prevalent and appealing in both art and science and nature? How does symmetry enable us to grapple with what are called monstrous numbers? And how might symmetry contribute to the elusive theory of everything? Joining me to discuss this are Ian Stewart, Professor of Mathematics at Warwick University, Marcus Uso-Toy, Professor of Mathematics at Oxford University, and Faye Dauke, reader in theoretical physics at Imperial College, London. Let's begin at the beginning, Fadaka. How would you define symmetry?

Symmetry is arguably the most important single concept in the development of fundamental physical theories that have taken place over the last 100 years or so. And roughly speaking, a physical system has a symmetry if there's some operation or transformation that you can do to that system, after which the way that that system behaves is exactly the same as if you had left it alone. So, for example, if we were to play a game of snooker on a snooker table in the middle of this room, then the way that the snooker balls roll around, the way that they interact with each other,

bounce off each other, that would be exactly the same if we were to move the table and play the game snooker over in a different corner of the room or put it in a different room of this building. The laws of physics, the laws of Newtonian mechanics are the same at different places. It's the same here in this room as in a different room. And there are other similar symmetries in the laws of physics. For example, if a law is the same today as it was yesterday as it was the day before, we say that that law is symmetric under the operation of. of time translation.

And another important symmetry is the symmetry that a physical law has if you put a system on a moving train. So we say that the law is symmetric under a particular kind of symmetry called boost symmetry if that law is the same on a moving train as it is on the stationary platform.

That's gone a rather a long way down the line at the start of the programme. In my sort of working of it out, We get to that at about half-b-bust-nine. In terms of symmetry being appealing to the human mind, most people listening, including me, very much. So I think symmetry is the old two hands, two eyes, two ears, two knees business. And the balancing up of things when you draw them

or when you see them in painting and stuff. Can you, Val-I, I quoted Valerie, where is it, the business is about it somehow present in the infrastructure of our intellect. So can you just talk about that perception of symmetry, why you think that we are attracted to symmetry? I would like to start there at the outside. In physics, I think that the attraction of symmetry in physical law is the attraction to the universal.

So if a physical law is symmetrical, then it means that it doesn't, it's independent of a certain happenstance. So to say that something is symmetric, is left-right symmetric, means that there's no difference between something and its mirror image. Yeah, but why do you think it's so appealing? Why do you think we're talking about it? Why do you think that symmetry matters so much? What's going on now?

It's mattered for, from we know, from early civilisation, if you come to a moment, why do you think it's, and can be used as part of the inner structure of the intellect? Is Valerie right about that? Just from personal experience, I certainly find symmetry appealing. and but as a scientist as a physicist, the question turns itself into why I find symmetry and physical laws attractive. And to me it's because of this universality.

So when a physical law obeys a symmetry principle, it means that it's more powerful than it would be otherwise. So if a law is symmetric under time-demeanor, translations, it means that it doesn't change from day to day. So we can say that that law has a more universal application. That's what's attractive about it for me as a physicist. Marcus Sissot, can you give us some examples of symmetry and how it worked in the natural world? Yes, I think, let me say for me, symmetry is more about, I think, starting with a physical

object, like a dice, and actually the very early examples of symmetrical objects you find in sort of neolithic cultures are sort of dice-looking objects. You find these stones carved into symmetrical shapes. And what makes symmetry in an object is, it's, again, I call them the magic trick moves. The things that you can do to an object, so if you take a cube-shaped dice, it's the things that you can do to that dice, pick it up, put it back down, and if the person has turned their eyes, they won't be able to tell a cube has moved. It looks exactly like it was at the beginning. So symmetry is the things that you can do to something which make it look the same.

Why I think it's so important is it denotes meaning symmetry. Something with symmetry is something special and it sticks out. So I think that's why we're genetically programmed in a way to see symmetry. Because in the jungle, for example, you've got all the chaos of the jungle. But if you see something with bilateral symmetry, with reflectional symmetry, it's probably an animal. It's probably either going to eat you or you could eat it. So I think we're naturally programmed to spot these things because they denote meaning. There's a nice sort of dialogue.

I think it's also about sort of a language in nature. If you take something like a bumblebee, bumblebee has incredibly bad vision. It can't judge distance. It's sort of colourblind. But the one thing it can pick out is objects with symmetry. And that's why the flower is chosen a symmetrical shape in order to attract the bee to it because it needs the bee for its genetic heritage as much as the bee needs the flower. the flower is chosen, the more symmetrical a flower is,

the more likely it is going to be visited by the bee, and the bee is just hypersensitive to symmetry. And I think that's the sort of beginning of why Valéry, for example, you know, it's the same thing. It denotes meaning it's something special, it's something we should take note of. Can you give us some examples in, because one of the interesting things about this,

is from the very beginning, from the beginning of ourselves making scratches on walls in the cave paintings, we see examples of symmetrical designs. That's right, yes. And that extends into music and poetry and the visual arts. Can you give us a few examples of that and say why you think it's in that area as well? Yes, I mean the very early cave paintings,

I mean, if you go to somewhere like Ireland, you see these swirls, symmetrical spirals, and they spiral them one way and then the other. And I think that is, because it isn't just some random sign. They're trying, this is the start of some of, attempt to communicate something with meaning either the cycles of the seasons are captured in these shapes. But it goes on then. I mean, I think all artists, they have an interesting sort of conflict with symmetry because they like symmetry, but they like also disrupting symmetry.

So, for example, if you go to the Ahambra, as you said, in the introduction, you know, that's, they were not allowed to depict images of animals and things with a soul. And so they went to some more geometric, symmetrical thing to express, sort of infinite complexity of God. through this idea of symmetry. And then you get it in music as well. I mean, I think variations on a theme is in a sense playing with symmetry. The musician is taking a theme and then doing things to it, which connect it to what it was before. And I think that's, I mean, symmetry, the word symmetry, simetros, it means the same measure in Greek.

And to start with symmetry as about equal measure, things happening the same towards the left as towards the right. but I think as we see the historical development of it, it becomes something much more than just measuring things with the same dimensions. Ian Stewart, do you think this goes back to, just to hold on for a moment, to the two hands, two eyes, to ears, business and what we are? Yes, I think a lot of this is built into the way the eye

and the works and the way the brain interprets what the eyes sees. And the physical symmetries that Pha was talking about affect the way the brain is structured. because if I look at an object, I have to recognise that object in different places, at different times, at different distances. So these are all symmetry operations in the physical world, and they're built into the structure of how the brain processes the images. And I think our aesthetic sense of symmetry is probably a sort of accidental byproduct of the actual structure of the brain and the way we see things. So, you know, it feels right. When you look at a symmetric object, you're seeing the same thing in the same thing.

lots and lots of places. It's been a subject thought about for a very long time. Can you tell us what the Babylonians did to it at first hour? And then we just go through the Babylonians, the Greeks and the Arabs, right? Your starter for about four minutes, really. Okay, the Babylonians, it's important to understand that the first several thousand years, probably the first 4,000 years of the history of this subject consists of mathematicians not knowing how to state what a symmetry is

and blundering into a correct and useful concept of symmetry that they can actually manipulate and think about, coming from a totally different direction, which is algebra solution of equations, rather dry sort of stuff sounds like. Actually, it's fascinating historically. And this is what the Babylonians did for us in about 2000 BC. There's lots of clay tablets have survived from the Babylonian period.

they have this lovely wedge-shaped cuneiform writing and people learnt to decipher them and among the million or so Babylonian clay tablets that now exist and have been found are a few hundred on mathematics and they range from exercise books at school for the trainee priests through to kind of research-level texts and tabulations

and a key point of Babylonian mathematics is they know how to solve quadratic equations. They know how to solve equations where you're told, in fact, a typical Babylonian question would be something like, I have a square. It's area plus twice its side equals 120. What is the side? And if you think about this, you think, oh, actually, if it's a 10 by 10 square, the area is 100, the side is 10, 120, that's the answer. but they had a system for finding it, and they had a rule, and they apply this rule, and they do it with complicated equations. So how does this relate to symmetry? In the mid-1800s, following this story of solving equations from the Babylonians,

mathematicians work out finally why the Babylonian method works. Right, but in between there, just briefly, did the Greeks and later the Arabs, especially this man Oma Kiyam, known more for his poetry than is, but did they add much to what the Babylonians have discovered? They added to what the Babylonians discovered. The Greeks were actually very interested in the regular solids, which are these very, very symmetric solids.

But at that time, nobody connected these subjects up. The Greeks were not very good at solving equations algebraically or numerically, but they introduced a geometric point of view. So Euclid systematizes geometry and some of Euclis successors were studying conic sections, curves you get by slicing a cone, parabola, ellipse, hyperbola. And they discovered you could solve some equations using these geometric curves. You draw various curves and see where they meet,

and then that tells you the answer to the problem. And they could solve cubic equations, which are more complicated than the Babylonians were working on. This is the next stage. As well as the unknown multiplied by itself, you have its cube. which is the unknown times the unknown times the unknown. Omar Kayam picks this up. The whole focus of mathematics moves away from the Greek, the Mediterranean world, and it moves more to the Far East and the Middle East, Arab, India, China.

And Omar Kayam solves all possible cubic equations by systematic methods using conic sections. But again, the symmetry there is still not a power. to anybody, either to our listeners or indeed to the mathematicians of the time. They're heading towards it, but they're heading towards it with their eyes closed. The reason they are able to solve these equations is because of hidden mathematical symmetries, which they're exploiting, even though they don't realize they're exploiting them. Is there a leap forward at the time of the Renaissance when they had these public competitions to see who could do and most quickly the solving of these equations?

The Renaissance is when the whole can of worms is opened, when some really big breakthroughs are made, and when they hit the big obstacle, which leads later to the discovery of what symmetry is and what it can do for you. Marcus isoto, so it seems that the Renaissance took forward, what Ian was talking about, very, I'm very grateful for you being so brief and succinct.

But then there was a halt for about 300 years. Why did they get stuck? What was they, where were there? Why did they appear, as it were, to get stuck in this quest? The point was we had some kind of formulas which had help us to solve quadratic equations, X to the squared, cubic, X to the cube, X to the 3, cortex, X of the 4. But when we came to things called quintix, which are equations with X to the fives in them, trying to solve what X is. We suddenly found that we couldn't find a formula. And so this was the big problem.

It was kind of like the Fermat's last theorem of its day. to try and either produce a formula to solve these quintics or, more interestingly, perhaps there wasn't such a formula. And it was perhaps one of our most romantic figures in the whole of mathematics, Evri's Galois, who died in a duel at the age of 20 in 1832 over love and politics. And the night before he was scribbling away,

putting the last details to his theory on actually why there was a fundamental problem when you went from the quartet to the quintic and it relates to a change in symmetries that happens underlying these equations. Now, I mean, why is there symmetry behind these equations? I think if you go back to the solving a quadratic equation, if I say x squared equals four, there are two solutions to this. X equals two, but x equals minus two is also a solution.

And that's like a mirror solution, so two and minus two. So you can already see a little bit of symmetry happening there just with solving a quadratic, you know, a square root can have a plus and a minus one. Well, things get a little bit, they're similar in higher sort of equations. So a quintic has five solutions. And what Galois started to do was to look at, what have you sort of swap those solutions around and play with them, a little bit like sides on a dice.

So these five solutions which solve this equation, he started to play around with what happens if you move them around and do they look the same or are they different. And amazingly, he developed a language to articulate this, which is a perfect language, and it's called group theory, to also articulate things about symmetry. And suddenly you could see that some of those wars in the Alhambra, which look incredibly different,

with this language, you could suddenly say, no, those have the same symmetries, although the pictures are extremely different. He is one of the most romantic figures in mathematics, isn't it? I mean, sort of solving this, the night before, and being killed in a love duel the next morning. No, no, you can ask for better. That's a PR for a mathematics.

Faye Daka, could you, Galois kickstarts what became group theory. Now, could you try to explain to persons like myself what group theory means and how it can be used? Group theory and symmetries are very tightly connected. And a group is a collection of operations that you can do to something. And so let's think of a square, for example. that square, there are several things that you can do to the square which will leave it the same. And for example, you can rotate it by 90 degrees.

That's one operation, a 90 degree rotation. 180 degree rotation is another operation. 270 degree rotation is another operation. Each of these operations leaves the square looking the same. There are also reflections that you can do of the square. it has four mirror lines of symmetry, and there are four reflections that you can perform. And together there are three rotations, four reflections,

and if we add the do-nothing operation, that makes eight operations that you can do to this square. And that collection of operations forms a group because you can do one operation followed by another, and the result of doing one operation and then another operation is one of the operations in your collection. There are other properties that this collection has

that make it into a group, but that's the most important one that you can do an operation and then do another one and the combination of the two gives you one of the operations that you already have in your collection. That's very clear,

but how did that move on to tell us more about symmetry? And this man, this man Galois seems to have exploded this into sort at that particular time. How does that move on from there? How do groups grow? How does group theory grow?

Well, different objects with different symmetries, have different groups associated with them. So the group of operations that for the square has eight elements in the collection and the way that they combine together is completely fixed. You do two operations, you get a third one and you know what that third one is.

But let's think of an equilateral triangle that will also have a collection of operations that you can do to it that leave it looking the same. And in that case, there are a few, six, I think, operations that you can do. So the group associated with the triangle is a different group. So you can characterize symmetries of these objects

by their groups, by this collection of operations that leave it the same. In sure, how does this liberate symmetry? It gives mathematicians, language for talking about it, for talking about when symmetries are the same kind of symmetry, different kinds of symmetry. If we just go back to Galois for a minute, the group of an equation of the fifth degree of a quintic equation is quite complicated. It has 120 different ways to shuffle the five solutions of the equation. And basically what Galois realized

was if there is a formula to solve the quintic equation, then this group of 120 different ways to shuffle the five roots must break up into five equal-sized pieces which fit together in a particularly nice mathematical way. And then just by looking at the group, you can't do that. This group does not, for group theoretic reasons, because of its own internal structure,

it does not break up in the way that a formula would demand. Therefore, no formula exists. It just can't be done. So at one stroke, studying the symmetries and looking at the internal, structure of this collection of transformations just knocks the whole mathematical problem on the head. So after that, people start saying, well, there's a 40-year pause

while the mathematical world wakes up to what Galois done because he hasn't published any of this. It just exists in manuscript form, nearly got lost. Well, he tried to publish it. He tried to publish it, and they wouldn't let him, yes. The academy kept on rejecting it. Yes, they didn't get it. No, they didn't just reject it because they didn't understand a word of it. They understood enough of it to realize he could do it better. And it got hung up in this horrible way. Anyway, but eventually the mathematical world woke up.

And people started saying, hey, these group things, they're really interesting. They're all over the place. Felix Klein, a German, essentially, look, all of geometry is really about group theory. You turn it on its head. Instead of saying, in Euclidean geometry, the transformations that leave things fixed are the rigid motions. You say, no, what's important is the group of rigid motions? and Euclidean geometry follows from that. And that's when they realise it's about symmetry.

So we're talking about group... Marcus Isoto, can you bring us back before... Let's stay in this area now, and it's hold on to your hats as far as I'm concerned, but still. Can you bring us back to the word symmetry? Is it being so elasticated from common usage that it has to be redefined for the rest of this discussion?

Yes, probably it does. I mean, I think that was... You know, we stumbled on this language by solving another problem, and we were really having a problem articulating what it means that something has symmetry. And I think the key point here is that the Gawar's breakthrough is to realize that it's the interactions

of all of these transformations that you can do to an object, these magic trick moves. That's the important thing. How they combine with each other to make new magic trick moves, that's the most important thing about group theory and not the individual symmetries. And so, for example, we had the equilateral triangle which has six symmetrical moves you can make on it, but also if you think of a starfish with six tentacles with a little sort of clockwise twist on them,

there are six rotations you can do to the starfish. Now, they both have six different symmetries to them, but this language of group theory can say those are different sorts of symmetries because of the interactions between each individual symmetry is fundamentally different. With the starfish, if you do one followed by another and do it in the other order, it doesn't matter. But in the equilateral triangle, it's very important which one you do first, and it changes the triangle completely if you alter the order. So I think that's the power of this language now.

You must think about the operations that you do to something which keep it the same as what a symmetry is. And I think that was a fundamental change. Then you start to see symmetry all over the place where not just in physical objects. Bell ringing, for example, is a good example of symmetry. You're changing the order of the bells, but it's still sort of four bells one after the other. But you can see that now as an example of symmetry. And that's why it's taking the power of group theories to take pictures into language. And suddenly with a language, you can say so much more and see symmetry in places that you never believed it was.

And do you want to come in on this guy? Well, I was also going to... Sorry. If I... Okay. One of the one of the things I think... Yeah, sorry. It is Gawar's discovery that there are sort of...

of atoms of symmetry. And the reason that the quintic cannot be solved is because the symmetrical object hiding behind the quintic is essentially something like an icosahedron, which is a 20-face thing with triangles. And that, the symmetries of that are somehow indivisible. You can't break them down into smaller symmetries. It's contrast with, for example, a 15-sided coin can actually be divided by a triangle and you get a pentagon. So a 15-sided coin. So a 15-sided coin, coin is divisible its symmetries because it's made out of a triangle and pentagon. But Gowah realized there are some really interesting symmetries like those of the Icosahedron, the rotations of the Icosahedron, you can't divide and it's somehow an atom of symmetry.

And this starts sort of a whole journey which went on for another 150 years culminating in the 1980s of trying to then say, well, can we produce a periodic table of symmetry? What are the indivisible symmetries from which all symmetries are built? Are we going to get there? But that was no, that was, that was fine. Just about at this moment in time, I think I understand what's going on. But then an atlas, there was a building up of an atlas, all the possible building.

Can we just talk about the Atlas before we move on? Would you like to talk about that? I'm not familiar. Well, that's sort of my, that was the periodic table of symmetry. Yeah, so we're done. So, yeah, that's why I thought it would be. This is the Atlas then, which, I mean, it's funny that we call it.

an Atlas because this book is a huge red book that was produced which lists all the building blocks of symmetry. And many mathematics, I've got one on my desk, which I consult all the time, because when I'm trying to understand something about symmetry, I can go to this, like the periodic table, which has hydrogen and oxygen, and I look inside this thing, and it gives me all the contours of these symmetrical objects. So the task was, in a sense, from Galois, Galois said, look, there are these indivisible things, go out and find them. And that's the journey that was made in group theory for the last 150 years. I want to come

to you a moment, but Ian can you very briefly say that we get some idea where this is taking us. What's it doing to mathematics? It's obviously adding to scholarship, but what is it enabling to happen so that people can get a fix? Or I can get a fix on what's happening outside

the intellectual absorption in this by a few, and it's still a very few extremely brilliant people? It's a very pivotal moment in the whole development of mathematics. It's the work of Galois and what surrounds it is the point at which mathematicians move from concrete problems, here is a question, solve this question, to very general abstract problems of what's the structure of this area, how do things fit together?

The whole of mathematics becomes much more general, much more abstract. Algebra is no longer just about calculations with numbers. It's about calculations with structures. And these key structure is this idea of a group. A group stops being a collection of symmetries and it actually becomes an object in its own right. So this Atlas, this catalogue, is a list of the basic objects and the mathematicians discover you can combine these objects together to get other groups and so forth. So there's now a calculus of groups which works in terms of the group as the basic

thing you're thinking about. And then the floodgates open and a whole pile of related developments occur And from about 1850 onwards, mathematics just changes its character completely. Faye Dauke, so group theory we've heard has become a tool that transforms mathematics. Symmetry in groups also transformed physics. Can you give us some idea as to how that came about? The symmetry principles are foundational in the developments of our modern theories in physics. and, for example, the symmetry that I mentioned before,

which is the symmetry that the laws of physics have, a law of physics can have, if you consider a system, putting it on a moving train, that symmetry was what led, it was holding fast to that symmetry, which was what led Einstein to develop special relativity. So he, he wanted the laws of physics to be symmetric

under this operation of putting the system on the train and he saw that the electromagnetic theory of Maxwell was not symmetric in that way, that if you have an electromagnetic system on a train, then it looks different. The laws that govern the behaviour of that system are different on a moving train. Unless you do something very well,

radical, which is completely reformulate your idea of physical space and time. So you have a choice. Eisen had a choice at that stage. Do we give up on the principle that the laws of physics should be symmetric under this operation of putting things on a moving train? Should we give up on that, or should we give up on our very deeply embedded notions of space and time? And he said, let's give up on our deeply embedded notions of space and time. Let's have a radical, formulation of what we understand by time. So for him he put the principle of the symmetric principle, the principle that the laws of physics should be symmetric under this operation.

He said that's more fundamental, that's most important. And by holding fast to that, that's how special relativity came about. And that transformed, can you just develop that a little more? Because it's fascinating. And when special relativity came about, it meant what? It implied and meant what? Special relativity means that space time itself has a particular, structure and it's no exaggeration to say that it underlies everything that we know about

everything we know about physical reality today at this stage all our best theories are informed by special relativity they're based on special relativity so gen relativity which is Einstein's theory of gravity assumes that that that special relativity holds locally and all our best theories of matter, that is the standard model of particle physics, the quantum theory of fundamental matter

is also founded on special relativity, on this principle that this symmetry principle that system should behave in the same way on a moving train, this so-called Lorentz symmetry. So it

underlies everything. Marcus Isota, can we talk at the main types of symmetry that physicists are interested in? The global symmetries, can you give us some examples of those or some instances of those? Well, I'm not a physicist,

I may not be, but there are very important building blocks called Lee groups, discovered by a Norwegian called Lee. And we find these actually are, they're the transformation groups in space that really describe what's going on under relativity. So there's some of the most important building blocks in this sort of atlas

that were discovered just at the end of the 19th century. I mean, for me also, I mean, I'd like to ask Faye this as a question actually, which is, I mean, I understood that also symmetry helps you to predict some of the fundamental particles as well. I think there's a story of Dirac where matter suddenly seems to have, the equations of matter is a bit like a quadratic equation

which has a sort of negative solution as well. And so he says, oh, okay, so there should be something called a bit like antimatter, sort of negative. And then, so symmetry helps you to actually find where these fundamental particles is. So that's, as I understand it, symmetry can help you to actually make sense of what looks like a whole menagerie of weird particles. Absolutely. I mean, that's a lovely example. And there are, in fact, several examples in the development of particle physics, where symmetry proposing a symmetry has led to predictions of new phenomena and new particles.

So, for example, the quark model of fundamental particles that was proposed by Gellman and others. And they suggested a particular symmetry between the quarks. And once you suggest this, once you propose this symmetry, the quarks come together and form. composite particles of corks, and there are certain patterns amongst those composite particles in particular. There's a group of eight particles, so this became known as the eightfold way. But some of the particles were not discovered yet, so there were gaps. So just as when Mendeleif put together the chemical elements in the periodic table, he was able to make predictions of new elements because there were gaps in the pattern. So the particle physicists of the time were able

to predict new particles because there were these missing particles which would have to complete the pattern if that symmetry were really true. And those particles were discovered. That was Nobel Prize winning. Yeah, I think that's amazing. The heart of nature, there's some huge symmetrical object, basically, which is plaining, or, you know, the building blocks of nature.

But it's both symmetrical and symmetry breaks down as well, Ian's here, doesn't it, at this level. Antimatter, I'm reading now from, probably from your notes, Antimatter should cancel out matter, but it doesn't, right? There's an idea of symmetry breaking, of things that are almost symmetric but not quite. So Marcus and Phae were talking about this symmetry between matter and antimatter that Dirac predicts, which is basically a kind of plus or minus sign in the equations. And in those equations, matter and antimatter are exact mirror images of each other.

But if you actually look at the universe we're in, it contains ordinary matter. and very little antimatter. You basically know whether there's any out there in space is not clear. We can make a little bit of antimatter in particle accelerators to see how it behaves. So how come, if when the universe first appeared, it had equal amounts of matter and antimatter, which would make it perfectly symmetric, they would just annihilate each other in a giant explosion,

and there'd be nothing left. But there's a little bit of matter left. So it would seem that early on there was slightly more matter than antimatter. It's not quite mirror symmetric in the same way that the human body, the human face is not perfectly left-right symmetric. So our universe was not perfectly matter-antimatter symmetric. And this is a big puzzle. And so you can approach this puzzle by understanding how symmetric systems can deviate very slightly from symmetry. Mathematically

the symmetries are still there but you've bent them a bit, you've disturbed them a little bit. Mathematicians are quite good at understanding what happens to things when you change them by very small amounts. So this is actually quite an attractive way of thinking about things. So we use the perfect symmetries of the ideal model to understand the imperfect broken symmetries

of the actual universe that we seem to be living in. Mark is Sertoy, we've dwelt a little while on physics, the unexpected behavior of symmetry. It raises new questions. Is the same happening, is something similar happening in contemporary mathematics? Well, yes, especially with the story of this Atlas. I mean, the periodic table culminates to this fantastically huge symmetrical object, sort of snowflake, which lives in several thousand dimensional space.

It's kind of an exceptional object, which doesn't fit into any other patterns, called the monster. and this monster, you know, mathematicians love these things because they're so kind of quirky and to play with but then suddenly we discovered some numbers associated to this monster which seemed to... What's the monster for and what's it describing? What's it all about this monster? Well, that's a good question. Yeah.

I mean, so this atlas is trying to piece together all... So it's a bit like herbium or something, a very large atom in the periodic table and so we didn't really think it had much connection with anything. It was sort of a very quirky object which mathematicians just love playing with. And then we spotted these numbers which were attached to the monster, sort of the dimensions it can live in, essentially. And we spotted, well, those numbers seem very similar to some numbers we'd seen in a completely different area of mathematics

in number theory called modular forms. And a guy called John Conway and Simon Norton in Cambridge saw these patterns and said, look, there has to be something which connects these two. And they christened this monstrous moonshine. So the interesting use of the... I mean, mathematicians love giving names to things, but moonshine is it sort of elicit alcohol sort of behind there.

But I think it's the idea is that there's some sort of sun shining on both of these things, sort of the monster symmetry and this sort of these numbers from modular forms. And the challenge was to try and find what this thing was. And it was a guy called Bortchards, who actually won a Fields Medal for discovering there was something in physics,

which helped us to explain the connections between these two. So everyone's got very excited about this monstrous symmetry because it does seem to also have connections with physics. There seems to be things called vertex operator algebra if one wants to know the particular thing, which seems to bind these three different areas together in a very deeply mysterious way, which I think we still don't truly understand.

So I think there's a real challenge out there still to understand what this monster is and why it's so important and why it seems to have something to do with nature as well. How would this play through in Stuart to... I know you've been talking about the real, real world, but the recognisable world that we all live in as well. How do these...

What you have been talking about, these new theories, these explosions of theories of symmetry, these explosions of groups and groupings and monsters, and how is that affecting... Well, the recognisable world, that we encounter every day? We've been talking about deep philosophical issues

on the frontiers of physics, but group theory and symmetry are also much more basic. They show up on the human level. For example, I said earlier, the way the brain, the visual cortex,

interprets signals from the eye, the structure of that is very, very symmetric. If you look at how the nerves are connected up to each other, you get a beautiful mathematical pattern, which has only been discovered in the last 10 years or so and you can understand a certain amount of how the eye recognises patterns in terms of the symmetries of the visual system.

I've been doing some work myself with an American colleague on the way our sense of balance works and that turns out to be symmetries and the amazing thing is on a group theoretic level it's the same symmetries as a cube as a dice which is what Marcus was talking about right at the beginning. you have a little mathematical cube in your head, but it's not shaped like a cube.

It's nerve cells that are wired up with the same symmetries as a cube. So this is the way that this abstract notion of symmetry actually comes right into things that we carry around inside our heads. And is it developing in your area, Fadark, is it developing new theories, is it, can you just give us an idea of what the present scene is there? Absolutely. So in fundamental physics,

the most important challenge that we face is to find a theory which incorporates and extends both gravity and quantum theory, so-called quantum gravity. And the historical precedent for using symmetry to develop new theories is very strong. I think people feel very powerfully that proposing new kinds of symmetry will lead us, will be able to make progress on these problems. whether that's true or not is something for the future to decide. So whether or not we need new symmetries, for example, there's a symmetry called supersymmetry, which is a symmetry between particles of matter and force carrier particles called fermions and bosons.

So many of my colleagues work on supersymmetric theories, hoping that this new symmetry will help us to further unify physics. My own work is not informed by that, kind of proposal of new cemeteries, but certainly I would want to be able to... I'm very sorry, honestly, my thought. We've only got seven seconds left, so thank you very much. Very darker Ian Stewart and Market of the Sauter. Thank you for listening. Next week, it's Greek and Roman love poetry. We hope you've enjoyed this Radio 4 podcast. You can find hundreds of other programmes about history, science and philosophy at bbc.com.com.uk forward slash radio four.