Finding Moonshine: A Mathematician's Journey Through Symmetry

from Imperial College London welcome to this special lecture vodcast on the 30th of May 1832 a peasant was walking on the way to market on the left bank in Paris and he heard a shot he ran towards the sound and he discovered on the ground a young man he'd been shot in the stomach and he was in great agony the boy who was taken to the coaching hospital where it's discovered his name was ever east Galois Galois wasn't quite sure what the jewel was about that morning it could have been over politics could have been over love it could have been the establishment trying to get rid of this rather awkward revolutionary who was kind of mixing things up a little bit but we're not too sure what the jewel was over but it's real passion actually wasn't revolutionary politics but mathematics he'd spent several years he was 20 years old when he was shot during this jewel he'd spent several years developing some new mathematics which he'd sent to the French Academy but no one really could understand what he was talking about so several of his manuscripts had got lost one that was read they really didn't understand what he was talking about and rejected by the establishment he turned instead to politics and that seems to mean what was involved in this jewel we were not too sure the next day he died in the hands of his brother and he said I need all the strength to be able to die at such a young age of 20 years old but the night before the jewel he realized it wasn't going to be the revolutionary politics that he'd be remembered for but his revolutionary mathematics and he realized he had to lay down his ideas that he developed over these last two years before he was killed so he spent the whole night writing away trying to develop these new ideas for this new language he'd developed maybe the reason he was such a bad shock it was so tired in the morning that he was they've been writing away and during that night he wrote probably one of the most important letters in the whole of mathematics explaining this new theory and what he developed is a new language to be able to understand the subject of symmetry now Galois R is a real here oh I think a mouse mark amongst most mathematicians but actually it wasn't when I was young it wasn't mathematics that I wanted to do I didn't see you know about these heroes or anything like that and when I was young there were a lot of young people here you always get asked what do you want to be when you grow up well once somebody asked me this I said I want to be a spy I wanted to be w17 this desire to be aspired being fueled by my mother's profession she before she gave birth to me and my sister and she worked the Foreign Office but when you give birth to children many years ago as a woman in the Foreign Office you get moved to sides and you're not allowed to be in the Foreign Office anymore so she retired but she was used to say that they'd let her keep the black gun that she always had in the Foreign Office oh my god and she said it was smuggled somewhere in the house so I put two and two together and realized my mom must have been a spy in the Foreign Office I I used to spend ages trying to search the house trying to find this gun this black gun hidden in the house and I could never find it I mean the obviously taught the art of concealment very well as well so I couldn't find this gun now I realized what the only way I was going to get a gun of my own was to become a spy and to join the Foreign Office so I asked my mum well what did you do well you need to learn lots of languages she said so I started signing up when I went secondary school for any language I could do so I signed up for French for German we did Latin at my comprehensive school there was a course on Russian on the BBC's so I started watching that as well and so my French teacher helped me with that but it was a disaster I couldn't say the word for hello in Russian it's so complicated how many ways it was the name of the course as well so I couldn't even say the name of the course my French and German started to get really frustrating there were all these exceptional verbs who didn't make any sense you had to learn the spellings of words and things like that it was just - I've got a terrible memory and I just I was getting so disillusioned they're not about 12 or 13 and my maths teacher at my school in the middle of a lesson he went to so toil want to see you after the class I don't gosh I'm in trouble now so at the end of the class he took me around the back of the maths block and I thought well I'm really in trouble now so and then he got out his cigar and he explained that the staff didn't like him smoking the cigar in his in the common room and this is where he used to smoke his cigar at breaktime and he said de Soto I think you should find out what mathematics is really about it's not really about all the sort of multiplication long division that we do in the classroom and he recommended a few books to me one of them was called the language of mathematics and it's not a very famous book I don't think but I I went with my father that next weekend up to Oxford I lived near Oxford and we went to Blackwell's it's wonderful there's a TARDIS of a bookshop this little front door you go in and then this huge books and in the basement there's something called a nineteen room which is where all the maths book were and I always like an Aladdin's Cave of books and so my dad found this one I sort of I sort of watched the students there they see me reading these books a bit like novels they were sort of against the side reading this I was totally fascinating what they were reading I pulled some of these books down just complete nonsense I couldn't make any sense of it at all but but I came to realize this was some sort of secret language a coded language and I became very intrigued especially when I got this book back home and I started to read about the language of mathematics and I realized as I read more and more of this book that the language that I wanted to learn when I grew up with this language of mathematics it seemed to be a language which described the natural world perfectly which didn't have exceptions there are very strange things which happen in this language but in some sense they all make perfect sense even though they are very strange and so I began to get addicted to this language and there was one topic in this book which particularly fascinated me which was this language which describes the world of symmetry and I began to realize that symmetry itself is a very important language in nature in fact it's so almost as soon as animals and plants started developing symmetry was the way that they started to communicate with each other symmetry is an indication of meaning in the natural world so for example if you take a bumblebee flying around the garden bumblebee has incredibly bad vision it's can't judge distances very well its world is sort of a world of black and white seem to be going around with a thick sand a pair of glasses where it can't really see things one thing it really can pick out is things with symmetry and it knows that something with symmetry is likely to be something with food and so conversely the flower as well who needs the bee in its natural development develops a symmetrical shape almost like a billboard saying come and visit me your sustenance here and in fact that's the more symmetrical or flower is it seems to be the more sweet the nectar is in the plant so swamis symmetry is there right down the bedrock of nature as a dialogue a language which helps the natural world to communicate to each other even with animals symmetry is incredibly important here are some pictures the same picture on the top what what's been done is that it's just taken a mirror image and reflected it so the pictures on the bottom have perfect mirror symmetry now most people will say that the bottom two pictures are more beautiful than the top two pictures you may be a quirky person who thinks the other way around but most people are drawn to the bottom two pictures and that's again because why do we like symmetry symmetry is a sign of very good genes if something is very symmetrical it's likely to have much better genes it's wasting it can spend its genetic time wasting time making symmetry so symmetry in the natural world is indicating somebody with very good genes which is why we're naturally drawn to them for example it's if you take battery farm hens the X that battery farm hens produce are generally a little bit asymmetrical and those that are free-range running round they produce much more symmetrical eggs because they've got the energy to be able to produce things with symmetry so it seems like symmetries that is being used by nature too as a way of picking out good good genes well I like this quote of Galileo's hoo he summed up something about the fact that symmetry and mathematics seems to be the best language to be able to describe the natural world he wrote the universe cannot be read until we have learnt the language and become familiar with the characters in which it is written it is written in a mathematical language and the letters are triangles circles and other geometric figures without which means it is humanly impossible to comprehend a single word and so it wasn't long before man was trying to understand this world of geometry and started to try and play around with what sort of symmetries can we find in nature and here are some Neolithic stones which state from about seven thousand seven thousand years ago they were found in northern Scotland and already you can see people carving these shapes into shapes with symmetry they're all very intrigued by you know what sort of possible symmetries can you have it's not quite clear what the significance of these stones is in Neolithic cultures it doesn't seem to be any associated with a game I mean they look a little bit like dice but they may have been very important as symbols for the clan symbols of power but we're not too sure what they are but you already you can see that they're experimenting with symmetry of course these do make very good dice these symmetrical shapes and the first dice that you find again date from a similar period this game was found in city code or in ancient Babylon and the dice are actually rather strange dice rather than the conventional six face dice that we play with these are little chattri he drew dice so they're they're triangular based pyramids with two little two of the corners marked white and two left black and you threw the dice the little pyramids and you count the number of white points pointing upwards and this is the way you count on these things so dice so obviously need symmetry because you want them to be very fair so these were already seven thousand years ago in order these shapes being used for dice so then you can ask well what's how many dice can you make what sort of shapes can make good dice and it was really the ancient Greeks who started that sort of more mathematical view of the world are trying to classify things they said okay can we work out what how many dice are possible and you find Plato recording the fact that there are in fact five dice five symmetrical shapes that you can carve out where all the faces are the same shape symmetrical shapes and Plato thought that these were so significant that he associated them with a sort of building blocks of nature so for example the tetrahedron which is up there at the top right hand corner a very pointy shape of course at the time they believed that the atoms of nature were earth wind fire and water and he felt that that one was associated with fire because it was a very pointy shape and you have the cube made out of squares six squares that was associated with Earth a next shape up with eight faces eight triangular faces can be put together into a perfectly symmetrical shape the octahedron and that one was associated with air and a twenty triangle as it was found could be pieced together to make something called an icosahedron which was associated with water it's the shape which looks most spherical and then Plato was left with one left over which is made out of Pentagon's 12 Pentagon's to make a dodecahedron and this he associated is used up four elements so this one he decided was the shape of the universe and it's very uncanny because although of course this looks a completely crazy theory now to associate these wonderful shapes with these things actually there's a lot of resonance with our real with the modern discoveries of what shape the universe is and what the world of the very small atomic structure looks like so it's believed actually that the universe may have some sort of flat shape to it in certain areas and beasts of dodecahedral in shape so although Plato could never have told that so yes if you look at the very sort of small structure of matter you find symmetry all over the place so for example crystals you find a lot of symmetry obviously in building out crystals so understanding symmetry will be key to understanding what possible crystals can exist even in the biological world simit tree can be very deadly many of the most deadly viruses are symmetrical in shape the herpes virus the AIDS virus are all symmetrical based quite often on the icosahedron partly this is due to the fact that the DNA in a virus is very small and so it needs a very efficient program to rebuild itself a virus wants to rebuild itself very quickly out of the genetic material and so these symmetrical shapes are very easy ways to rebuild a shape because there's just sort of a very simple rule to put these shapes together so we find symmetry all over the place actually in the very world of the very small if you look at the very fundamental particles there's we've discovered all these weird and wonderful things gluons leptons protons and neutrons well they look like a complete mess but physicists have been able to make sense of this whole menagerie of fundamental particles in a sense by identifying them as all facets of a symmetrical shape in a very high dimensional space so symmetry is really at the heart of the way science is put together now science of course symmetry is very important to science but it's actually very important to the artists as well and I think one of the artists one of the arts which has most resonance with mathematics is of course music music a mass has always sort of put lumped together and you find a lot of symmetry being used by composers because symmetry is about sort of a connection between different parts of the same object and embark probably is the master at using mathematics in his music his student mitla used to say that music is the process of sounding mathematics and this piece in particular the Goldberg Variations I think is sort of the height of symmetry in music so here's one of the movements of the Goldberg Variations the Goldberg Variations consists of 32 movements starts with an aria and ends with an aria the arias are the same and they sort of connected up so the piece is almost like a circle in fact if you go to the piece in the middle the sixteenth movements bark causing an overture so it's always not clear where is the beginning of this circle so you've got a circle already in these 32 movements every third movement is a cannon a cannon is like those songs for arishok aware somebody starts the same tune a little bit later so the first cannon has the the theme and then it's repeated a second time and the two run together but each time you hear the cannon a next time the second voice moves up a tone so you hear the first theme and then the second theme is one note higher and then when you hear it again it's two notes higher three notes higher until it goes all the way around and you hear it in the eighth cannon where it joins up because you've got an octave so suddenly it's an octave higher which sounds again like a circle completing itself so in since you can see the Goldberg Variations there's a beautiful example of a musical version of something we call a Taurus so this is what we've got here a Taurus is a circles worth of circles so you can almost think that this is this symmetrical shape the Taurus it is the structure of the Goldberg Variations and the rhythm section as well if you look at the way the rhythm of each of the cannons in this piece is constructed actually Marcus made sure that he's got every single different rhythm structure the beat is divided into either two two quavers three triplets all four semi quavers and then the beat is divided into two three or four beats and bar goes through all the different possible variations on this sort of a combination lock here the symmetries of this combination log in order to ensure that he's covered all his bases there's a huge amount of symmetry in this piece and I mean bark is at that time was the sort of master asymmetry but there's a it's a modern composers really who I think have really taken the idea of symmetry in one step further than the simple use is that Barco is making as anarchists may compose a piece called Nomis alpha and i want you to listen to this piece and try and guess what symmetrical shape sown arcus is trying to represent in this piece so here it is are you getting a shape in your mind well that wasn't practicum was this stock in there yes quite hard to hear the cube I I agree but what's anarchist was doing was to put on the edge of the cube different musical ideas for the solo cello and he would have a again a sort of a theme and he would bury the theme by permuting by moving the cube around and all of these musical ideas would move around and somehow the combinations would reflect the symmetries in the cube he couldn't move any of all of the eight corners into any place because there was some rigidity in it so the variations are being put into this piece of music are somehow meant to reflect the symmetries that hidden inside this cube as an artist was in fact an architect as well so there was a lot of thought very geometrically in his music but I think it's pretty hard to hear the cube in there I must admit it Sam but I actually artists have a little bit of a difficult time with symmetry they they kind of like it and they don't like it because sometimes it's too prescriptive and here's Thomas Mann talking about symmetry in the magic mountain and he's describing a character is describing a snowflake and he describes it that he shuddered if it's perfect perfect precision found it deftly the very marrow of death and for a lot of artists symmetry is almost too constraining and it's a symbol of stagnation and death and so very often they want to break out of that symmetry but of course you have to have the symmetry there in order to be able to break out of it and say something different I like this quote especially from the Japanese essays in idleness which is describing symmetry in architecture it says in everything uniformity is undesirable leaving something incomplete makes it interesting gives one the feeling that there is room for growth even when building the Imperial Palace they always leave one place unfinished and in fact I described in the book a trip I made to a temple in nikko a conference that I was at in Japan and we went to the group of mathematicians and when I enter the temple there are eight columns in the entrance for the temple and they all have this beautiful symmetrical pattern all but then I noticed that one of the columns at the back had been turned upside down after this is very strange this beautiful symmetry but one at the back had been turned upside down and asked my Japanese colleague you know was this some sort of mistake and then he explained this quote from the Japanese essays in idleness that they don't like perfection there has to be a little bit destroyed somewhere and for example Arabic carpet Weaver's also will weave a little bit of imperfection into the corner of the carpet in their case they don't want to sort of simulate God God is the only one who can prove produce perfect perfection and again if you look at something like Bach's Goldberg Variations when you get to the tenth Canon you're sort of expecting all the symmetries build up you think you know what is going to happen but then bark produces what he calls a musical joke Accord Libet and completely misses with what you're expecting but you'll almost reinforces how much symmetry has gone on before when you hear this thing which clashes with all of all your expectations so artists are very it's very important for them to have symmetry and it sense to break out of it and say something different well this temple in nikko certainly had a lot of symmetry around it and buildings architects have used symmetry very often is a mark of saying something you know this is important for example the pyramids where the Pharaohs are buried underneath they they're saying that you know you should take notice of this this is important in Paris at the time actually the Galois got killed they already had this idea to produce a perfect sphere in Paris they felt a spherical building represented the ideals of the revolution perfectly equality a gala Tay at the heart of society and so they had this idea for building a spherical building it only got realized 200 years later if you go up to LabVIEW let's where are the science unit is where the science museum is you find this beautiful silver globe it almost looks like it's landed from outer space and down underneath you can go inside it is in fact so an IMAX cinema not quite what the French Revolution I think had in mind for their sphere about Sam I never mind it's still worth visiting also I like these ones these are houses in Ramat Gan in Israel promote in Israel their dodecahedron is essentially put together in this kind of bonus like this beehive but but but the this house is terrible to buy furniture for goodbye this is a weird sort of shape so but I think of all the buildings which really where symmetry has been celebrated if I was cast out into a building that I had to spend the rest of my life in it wouldn't be any of these it would have course be the Alhambra in Granada which is somehow the Palace of symmetry and symmetry all over this place the Moors who built this Muslim law denied them any pictures of things with Souls so they had to find different ways to express their their artistry and of course they went for a lot of geometric figures you find symmetry of course being said you can see a lot of symmetry with the amount of water that's there all the reflections that are going on with the the pools in the Alhambra but it's really on the walls where the symmetry is celebrated too it's to the extreme the walls seem to explore all the different sort of symmetries it could be possible to put on this wall there's so many different designs different ways of putting together these different tiles and things but um I think the oh amber a really raises the questions about well what is symmetry exactly how can we say whether any of these walls are the same symmetries have the same symmetries what does it mean to say that these walls have symmetries and I described a trip that I made till the Alhambra in the book with my son who was nine at the time and I was very intrigued to see what he thought about the walls in this in this palace whether he thought there was symmetry what were the symmetry is here um so example the first entrance is when you go into the Alhambra the the first place you meet is this one here and I asked him you know just this house cemeteries this doesn't have any symmetry at all I said what you mean well it's a there's no reflectional symmetry in here that was his first impression that symmetry is about reflectional symmetry and I think that's most people's first impression that something symmetrical this has left and right if it all matches up well I said yeah it's not got any reflectional symmetry but what about rotational symmetry and he started to see that yes you could lift the tiles up and somehow rotate them and they could all fit down on top of each other but to really to be able to articulate well okay are any of these symmetries then the same the pictures are very different how do you do that and well it took us thousands of years really be able to find this language to say what is symmetry to be able to tell where the two symmetries are the same or not and that is what this guy out every scowl are did by the age of 20 he developed a language to be able to say exactly what symmetry is here's one of the one of the scribblings and the letter that he wrote the night before actually are not true I think this might even be one of the letters he sent to the Academy so you can see why they were having trouble with actually penetrating exactly what fir Galois was hoping to express but what Galois R understood was that symmetry is not about death and stagnation it's actually a lot about movement and action and energy because what he understood was how can you describe what the symmetries of these objects were and the way I try to explain to my son when we were walking around the Alhambra is its symmetry is a bit like the magic trick moves that's how I described them you're given a shape and if you draw an outline round that shape and then the symmetries are the ways you can pick that shape up put it back down it's an ID it's out line so you ask the person to turn away you pick the shape up turn it down put it back down again and then you ask them to look back and they can't tell whether the picture is moved or not it's all the motions that you can do to something which make it look like the same before you moved it so there you ask person to look away and then you'd make some move on these things so yeah I'm going to move to this overhead projector here so for example so what are the symmetries of these two shapes let me try and explain to you so so you take this starfish here with its six points and you make an outline of that starfish that blue want outline underneath here and then the symmetries of this shape and this is what I learnt in this book the language of mathematics the symmetries are the things that you can do to this shape where you pick up the shape and you move it some hill and place it back down inside its outline so it looks like it hasn't moved so now I've made a turn at one sixth of a turn and I can make a turn of two sixth of a turn three six four six five six or in fact I could have picked the thing up and left it where it was put it back down the same so so Galois would have recognized that this thing has six symmetries there are six things I can do to this shape where I pick it up and I can put it down back down inside its outline now I can move to the triangle how many symmetries does this object have well again here's the shape and now I'm going to make an outline of the shape underneath it oops let's get my letters around the right way and then how many things can I do what how many things actions can I make on this shape where I can pick it up and put it down back inside its outline well this time it's got reflectional symmetry unlike the starfish so I can pick it up and turn it over and this point at the top has stayed the same but these two swapped over as obviously I can there are three of those I can fix the point why turn that over or I could pick the point is it turn that over so there are three symmetries they're called reflectional symmetry z' or i can rotate this shape so i can rotate it a third of a turn in if it's down inside its outlying or 2/3 of a turn and then there's a sixth symmetry which I was when it there's a kid I always found this rather curious the symmetry I can pick it up and put it back down again where it was the start well that's a magic trick move you know it's still gone back in the same so actually these two shapes both have six trees so maybe we should say that they're the same symmetries they have the same number of symmetries but of course we have some feigning light no that's not right because they seem to be sort of different summer rotation summer reflections for how can we articulate really that these are different symmetrical objects well I'm going to set you a little exercise to see how good you are at at counting magic trick moves so these are quite easy shapes there are six different things I could do to this shape so the symmetries for example of a rubik's cube are all the things that I can do the Rubik's Cube which make it look still like a cube okay so the colors you can all the colors because basically the colors are going to look different but how many ways can I rate we arrange all of these cubes on the face such that the thing still looks like a cube okay I want you to think about that you can write down a number I'm going to give you a little prize I'm going to give a prize to the person who gets nearest to the number of symmetries well this little kids here is already I'm going to give you a little time okay so you can have a little guess at that one all right I'll come back and then will will I'll do a little competition see who's got closest so if you want to take part write down a number now and I will come back to that so the symmetries of this shape or the number of things you can do to the shape which which keep it looking like a cube okay and I'm going to give actually a symmetrical shape to the person who gets closest but how can we articulate that fact that these symmetries of these the starfish and the triangle seem to be different well Galois already had the first ideas for this language but it was really it took until it's about fifty years later when arthur cayley a british mathematician began to write down a language which would really articulate somehow to tell that these two shapes are different in their symmetries and this language the key is not only what the magic trix moves are you can make on a shape but how they interact with each other when you do one symmetry followed by another what effects that have on the shape so it's the interactions of all these symmetries which really brings out the true nature of the symmetries of one of these objects so for example here's a little bit of mathematics for you this is a description of how the symmetries of these different shapes interact with each other so I've given names to the rotations of this starfish the six pointed star fish so this is leave it where it is a B is rotated by sixth of a turn C is 2/6 D 3/6 4/6 F five six and this table records the way that these symmetries interact if I do a turn of 1/6 of a turn followed by three sixths of a turn that leaves the shape twisted by four sixth of a turn so if I do be followed by D it gives me e so this table Katie realized was somehow showing you how the symmetry is interacted with each other and it was this that was going to give the key to telling these symmetries apart because now if I look at the triangle I've got x y&z which are the reflections in the X point the Y point and the Z point and then U and V are the two rotations and the way these symmetries of the triangle interact with each other are incredibly different to the symmetries of the six pointed star fish if you look at this table the interactions of this table there's a symmetry down this this middle line here and that's because it doesn't matter what order you do rotations in if I rotate by a third of a turn and then a sixth of a turn that's the same as rotating by six and then a third but with the triangle if you do reflections followed by a rotation the triangle ends up different if you do a rotation followed by a reflection and this is indicated by the fact that there's no symmetry along this diagonal line in the table of the interactions of the symmetries of the triangle so Kaylee's language is development of Galois language enables us to say well the symmetries of these two shapes are different because the way they interact with each other is different and Cayley wrote down some laws for the tables for these symmetrical objects let's make like filling out a Sudoku there are certain things if you look in each row in each column each each symmetry only appears once and once only so there are certain rules for the way these tables and interactions of symmetry can behave and so if you even has six symmetries these are the only two tables possible so using this language we can say there are only two different objects with six symmetries and the shape may look very different but if it's got six symmetries it has to be have the same symmetries as one of these two shapes and this idea is already said of moving the subject of symmetry on is a bit like the concept of number I've got two tables here and two chairs now the tables and the chairs are very different but the Turnus of the thing we identifies the same somehow the number of the number here is the same although the objects are very different so what you need to think of is symmetry is very similar symmetry is sort of an abstract concept which underlies these geometric figures so the geometry of the figure may be very different but the symmetries may be the same so for example if you go back to the Alhambra and you go to some of the walls here are two walls who look very very different one with this beautiful triangle with a sort of twist on it and then it's amazing sort of six pointed star here the symmetries of these two walls are actually the same because the symmetries remember the magic trick moves so that's all the different ways I can lift the tiles turn them around and put the tiles back down again in sound inside an outline well there are the sort of symmetries that you can do there's a symmetry where you can rotate by six of a turn around the points with the six on it then a third of a turn which is a turn around the middle of your piece and then there's actually another symmetry was a rotation of a half of a turn 180 degrees which is half if you rotate at a point halfway along the edge of one of these triangles and what have you described the way the symmetries of the little twisted triangles work is exactly the way as the symmetries of this second wall behave so although the pictures are very different we can say the symmetries of these objects are now the same using this language so here's another example this symmetry is called four four two and nothing to do with football but it means that there are two points where there's a rotation of a quarter return in there are different walls and one point where there's a rotation of half a turn and although the pictures again look very different the rotations behave in exactly the same way in both of these pictures so all of these three walls are exactly the example of the same symmetry something called 4-4-2 we're using this language we can now actually say well what are the limitations on this wall how many symmetries can you have on the walls in the Alhambra well it turns out there are actually only 17 different symmetries that you can have on the wall in the Alhambra anybody who tried to produce an eighteenth one will be unsuccessful you'd always be able to show that that wall must be one of the of these seventeen different sorts of symmetry and it's the power of this language of Galois z-- and Kaylie's which says that that we can say that what the limitations are there are only seventeen different ways that these symmetries can interact with each other so and we've been able to classify these things and now we can take a symmetry on the wall and we can identify the one of these seventeen so that's the power of this language it shows us what's possible but also shows us four limitations and on this journey around the Alhambra with my son what we try to do is to try and discover whether they done in fact discovered all seventeen different sorts of symmetry in the walls in the Alhambra of course they didn't have the language at that time to really understand what they were doing but they were so invented that they produced nearly all seventeen you had to cheat with one of them but so if you go to the Seville or the Alcazar you can find the 17th one there well the other wonderful thing that Galois did he produced this language to be able to say when symmetries are the same or not to be able to just produce limitations of symmetry but he also made the most amazing discovery that symmetries also have building blocks that symmetries can be broken down in kind of into indivisible symmetries that in fact of the heart of symmetry is something like the periodic table in chemistry and so suddenly it opened up the idea well now can we classify what the building blocks of symmetry are what do they look like what are the building blocks let me give you an example if I take a 15 sided figure okay this is not an indivisible symmetry why because the fifteen sided figure the rotations of a 15 sided figure can be reproduced by using the rotations of a Pentagon sitting inside and a triangle sitting inside so any rotation of a 15 sided figure I can do by doing a rotation of a Pentagon and then doing a rotation of a triangle for example and there this is hiding behind here it's the fact that 15 is equal to 3 times 5 so primes which of the building blocks of numbers are also the key to the first building blocks of symmetry so for example how can I move point A to point B which is a 15th of a turn using the Pentagon and the triangle well what I do is to move point a all the way around to Point C by doing two turns of the Pentagon and then I use the triangle to pull Point C back up to point a by doing a third in the turn the other way and the the number arithmetic behind this is the fact that one fifteenth of a turn is equal to 2/5 of a turn minus a third of a turn so all the rotations with a 15 sided figure can be done by using the building blocks to the Pentagon and the triangle so the prime sided figures are the first building blocks in this periodic table of symmetry but there are more and this is what Galois discovered Galois discovered that there are some shapes that might have a non prime number certain number of symmetries but which cannot be broken down into smaller symmetries for example if you take the icosahedron which is the heart of the herpes virus and this is a chocolate box that Asscher design and the psychic icosahedron I sure did a very clever trick what he did was to put a little twist on the on the starfish so he's destroyed the reflectional symmetry in his chocolate box was actually designed for a chocolate 50th anniversary of a chocolate manufacturer in Holland I think it was but the symmetry and a you can see in an example if you if your inner ever in Den Haag then go to the Escher Museum there's an example of this box a really beautiful tin box but a Galois realized this is rotations of this object there are 60 rotations you can make of this shape 60 is an incredibly divisible number but the symmetries are as indivisible as if 60 was a prime number you can't divide this shape by another shape and get some sake with symmetry you might think well surely there's a the symmetries of a Pentagon sitting inside here after all you can see the rotations of that starfish yes there are they sit there as a subgroup of symmetries but if you try and divide by that object you don't get left with anything that makes any sense you've just just not divisible and so this is the first example of an indivisible symmetry in this periodic table that mathematicians have been producing which is not one of these prime number shapes so the task was on once Galois had discovered this to try and classify what all the building blocks are what sided have happen is that people found that these oh let me just tell you that I'm in this symmetry of anybody I'm a big football fan and I'm missing Arsenal playing Blackburn tonight to come and talk to you but the symmetries for example the rotations of the symmetries of a football are actually the same as the rotations of an icosahedron and so the symmetries the rotations of a football are in fact the first building block in the classification or in the periodic table of symmetry which are not a prime number shape now this is an example actually of an infinite family because the symmetries of this are the same as the symmetries of a pack of cards and where you have five cards in the pack now where are the symmetries in there well again it's the magic trick moves you think of all the shuffles so you can make up a pack of five cards I wish the pack comes back to the original place it was so you can think of all the way of shuffling packs of cards when it's not quite indivisible but it almost is now this shape kind of runs out so you know what's the next shape after that one but a pack of cards will wide not take six cards or seven cards or eight cards and what's Galois discovered is that the symmetries of a pack of cards with any number of cards in is actually also an indivisible symmetry so he produces infinite family where you can have 52 cards 53 cards and it's symmetries of the shuffle of a pack of cards a part of this periodic table of symmetry now mathematics is about a subject of looking for patterns and so we love putting things into these wonderful patterns and so we produce more and more families of these indivisible symmetries over the decades after Galois some more geometric beautiful structure of this heroic table seems to be producing except there were five that didn't sort of fit into any pattern at all discovered by a French mathematical called Macchio now just saw with people just said okay let's just ignore these five they were rather distressing but these lovely patterns and things that's ignore these some perhaps they'll make sense it's a part of an infinite family later on that was all well and fine the the periodic table of symmetry seemed to be settling down and then in 1965 a Yugoslavian mathematician called yangko discovered another sixth rather strange indivisible symmetry which also didn't seem to fit into any pattern at all and after that one another one appeared and another one and there seem to be these spiky exceptional symmetry indivisible symmetries which weren't fitting any to any patterns at all they produce seven of these things eight 17 20 of these things started appearing and people go began to get worried what is this periodic table of symmetry going to look like it seems to be going on forever and eventually in the early 80s an extremely large symmetrical object was discovered which is indivisible but doesn't seem to fit into any pattern at all this is how many symmetries it has there's a lot of symmetries more than there are atoms in the Sun it doesn't seem to fit into any other family and you only start to see this symmetrical object when you move to one hundred and ninety six thousand eight hundred and eighty three dimensional space that's the first time you see this shape okay it's kind of like a snowflake with Sir very strange symmetries which this snowflake only appears in this dimensional space now you might say well how on earth can you to wrap six thousand eight hundred eighty three dimensional space well let me show you how mathematicians do that one minute on high dimensional space the point is again it's about producing language we produce a language to be able to see new symmetries where we didn't think there were any well language is also at the heart of seeing high dimensional shapes Descartes produce this wonderful way of changing geometry into numbers for example you take coordinates so a two-dimensional shape you have a sort of vertical and a horizontal direction and you can produce coordinates for these things so for example I can describe a cube by the points in Cartesian geometry so the point at 0 0 1 0 0 1 and 1 1 okay that describes the square for me the cube similarly I can move in three different directions so here we start at 0 0 we can move one in sort of vertical plane horizontal we can move up 1 and so this Q can be translated into the eight points described by these numbers well what about four dimensions well of course I can't draw anything now I can still write down the numbers they don't run out I can write down four numbers with zeros and ones and different places you'll have how many we have 16 16 points so cube in four dimensions has 16 points described by 0 0 0 0 1 0 0 0 and I can talk about the symmetries of these things because these points are all connected to each other an edge you can describe one an edges and so I can describe symmetries in these higher dimensional shapes for dimensions why out five dimensions just have five points with zeros and ones I couldn't talk about a Cuban five done inches so the point is that what we're mathematicians say why I'm seeing a thing in five dimensions not seeing it physically but they've changed it into numbers where the numbers help you to explore these things you can see shadows of these things so here's a shadow of a four-dimensional cube in Paris the arch at ladee performance have you been to the arch its impact to a shadow four dimensional cube I'm not sure how many prisons no that's per term if you think about how an artist draws a cube on a two-dimensional page they draw a square inside a square and join up the edges may gives you a sense of depth well this is what the architect at laude force has done it's a cube inside a cube and in fact you can count the 16 points on this hypercube they're all here there are eight in the middle and eight on the outside so we can draw pictures of these things now okay four dimensions may be fine but what about is 196,000 880 three dimensional space where there are people who can think in these sort of dimensions and be able to play with these objects and see that these things have symmetries which cannot be divided so we had this wonderful classification of all the building blocks of symmetry and eventually at the beginning of the 80s it was discovered that there were going to be no more there was lovely the internet families and then 26 rather exceptional things and the one being this thing called the monster because it had so many symmetries and now one of the experts in the world of symmetry is a guy called professor John Conway who was at Cambridge at the time when this classification finished and he decided we should produce a sort of a periodic table of all of these symmetries and he started writing down all of these symmetries in something called the Atlas and I brought a copy along here this this is one of the most important books in mathematics and this contains the periodic table of symmetry so it's basically a list of all the building blocks are of symmetry in fact so the things so symmetrical if you look at their names the only you could only get on this list of names if you had six letters in your surname and two initials so when I first first went up to Cambridge to see whether to do my PhD there John Conway said are first asked me well what's your name and when I say that's hopeless I'm gonna have to drop the due and definitely drop an initial and so and and also as when they join the group do you see see see NP w so I was going to also have to change my s to Z or something to I'll get on this list so these guys are obsessed with symmetry and and so this contains now the building blocks of symmetry we have a complete periodic table of the building blocks of symmetry but the task now is to see what we can build out of these symmetrical objects in fact there's some very strange things on this number one hundred and eighty six thousand one hundred and ninety six thousand eight hundred eighty-three isn't some random number but seems to have connections with very deep things in number theory and this is what the moonshine and the type of the book is trying to understand why this very strange object which exists only in this dimensions this is the first dimension it exists why does this number have such strange relationship with other things in mathematics and physics so that's still we still got to try and sort this out erm and there's been some progress made but there's still a lot and really not understood about the monster what I'm involved in is actually my research I I'm I am in fact a group theorist at heart I do a lot of number theory as well but what my research is dedicated to is trying to see what I can build what symmetrical objects can you build out of the building blocks in this table and so I've been producing bit like molecules this is like the periodic table with all the atoms in and you say what molecules can I build it can't be a water to hydrogen in an oxygen so I'm trying to see how are you peace of these building blocks together and in fact some the prize for the first person who gets closest to the number of symmetries of the Rubik's Cube I'm going to name a symmetrical object that I built earlier on so here it is it's waiting for your name to be put in there okay so I've got quite a few of these things there they're my proudest moment was inventing these groups because they have connections with symmetry with another area of mathematics called elliptic curve some so I'm going to dedicate this group to one of you okay and your name will be on this group so I want you if you want to play this game you have to stand up if you think you know how many symmetries there are in the Rubik's Cube okay so if you want to play and try and get your name on this thing and then please stand up okay you had to have written the number down I want you all to play fairly okay okay so if you want to write you have you written your number down I need it in decimal notation no factorials please yeah you have to make sure that what yeah so you have to calculate the factorial now okay so okay so stand up if you want to play okay the only a few of you wanted well you're too scared to stand up hey Greg very good yeah all you have to do is to write down a number okay what can be more difficult than that just write down any number and I have a guess at how many symmetries do you think there are of the Rubik's Cube okay here are my players on it okay great okay now count the number of digits in your number two okay that's interesting well if you've got less than ten digits sit down okay right how we got me we got one two three four five six okay if you've got more than thirty digits sit down okay if you've got more than 15 digits keep standing okay if you've got more than 20 digits keep standing okay if you got northern oh we've got a winner then all right okay so how many digits is your number have how many 22 digits two digits so you mean like you thought how many symmetries do you think it had 24 no no okay so I mean okay so my two yeah you can stand up again the two that were had more than 15 digits so it's a huge number of symmetries they're a huge number of things you're absolutely right the number of rotations that you can do the cube but I'm talking about moving the sides as well like in a Rubik's cubes there are 24 rotations that you can make to a cube but now you also have to move the sides about okay so what did I get up to more than more than 18 digits okay you got more than 18 digits how many in yours 20 okay there are in fact 24 digits in this number so but what's your what's your name max what's your surname I think you better have your surname on this there's a hard one is it okay all right let me find my my pin where was my pin gone oh no here we go now I need a okay so spell your surname for me mi mi TR yep okay H I in there you are the MIT rock in group Wow very good so thank you very much well done so there's a new group of symmetries that was built out of the symmetries of the building blocks in this table and now named after the person who won the prize so here we are the rubik's cube how many does it have well it actually has this many symmetries so a huge number so if you light lay these cubes out in line they would actually go from one end of the universe and the other all the different possibilities so huge number of symmetries that you can make of this cube amazingly the ideal company toy company who made this cube stated on the package of the original Rubik's cubes that there were more than three billion possible states which somebody rather discrete nicely described as Mike McDonald saying and now they'd actually sold more than 120 hamburgers there's a total in comprehension about large numbers and the Rubik's Cube you can also break down it's not an indivisible symmetry it's actually built out of symmetries of these shuffles of packs of cards so if you take eight cards pack of twelve cards rotations of triangles and 26 mirrors they're the building blocks which make up the rotations of this rubik's cube so every symmetrical object can be broken down into the building blocks that are in that table now I'm going to come back to this quote which I really love of the Japanese essays in idleness because some the wonderful thing about mathematics they're incredibly large number of problems which we can't answer we still really don't understand the molecules that we can build from this table which is why my research is dedicated to creating things like this and it's a bit like the quote from this Japanese essays and idleness said mathematics is really and everything uniformity is undesirable we quite like subjects where there are these exceptional things like the monster leaving something incomplete makes it interesting the fact there is still unsolved problems in mathematics is what keeps mathematics a living subject and it gives one the feeling that there is room for growth and this certainly is then I hope the young people in the audience will join the search but try and understand what symmetries we can build from this table even when building the Imperial Palace they always leave one place unfinished there is certainly more than one place unfinished in my subject ok thank you very much you've been listening to a special lecture podcast from Imperial College London

Channel: Imperial College London

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Id: pmbxM4IcDtA

Length: 56min 1sec (3361 seconds)

Published: Thu Jun 23 2011