Symmetry, reality's riddle - Marcus du Sautoy

on the 30th of May 1832 a gunshot was heard ringing out across the 13th arrondissement in Paris peasant who was walking to market that morning ran towards where the gunshot had come from and found a young man writhing in agony on the floor clearly shot by a dueling wound the young man's name was ever East galois he was a well-known revolutionary in Paris at the time Galois was taken to the local hospital where he died the next day in the arms of his brother and the last words he said to his brother where don't cry for me Alfred I need all the courage I can muster to die at the age of 20 that wasn't in fact revolutionary politics for which Galois was famous but a few years earlier while still at school he'd actually cracked one of the big mathematical problems at the time and he wrote to the academicians in Paris trying to explain his theory but the academicians couldn't understand anything that he wrote this is how he wrote most of his mathematics so the night before that Joule he realized that this possibly is his last chance to try and explain his great breakthrough so he stayed up the whole night writing away trying to explain his ideas and as the dawn came up and he went to meet his destiny he left this pile of papers on the table for the next generation maybe the fact these stayed up all night doing mathematics was the fact that he was such a bad shot that morning I got killed and but contained inside those documents was a new language a language to understand whether the most fundamental concepts of science namely symmetry now symmetry is almost nature's language it helps us to understand so many different bits of the scientific world for example molecular structure what crystals are possible we can understand through the mathematics of symmetry in microbiology you really don't want to get a symmetrical object because they're generally rather nasty the swine flu virus at the moment is a symmetrical object and it uses the efficiency of symmetry to be able to propagate itself so well but a larger-scale of biology actually symmetry is very important because actually communicates genetic information I've taken two pictures here and I've made them artificially symmetrical and if I ask you which of these you find more beautiful you'll probably be drawn to the lower two because it's hard to make symmetry and if you can make yourself symmetrical you're sending out a sign that you've got good genes you've got a good upbringing and therefore you'll make a good mate so symmetry is a language which can help to communicate genetic information symmetry can also help us to explain what's happening in the Large Hadron Collider in CERN or what's not happening in the Large Hadron Collider in CERN to be able to make predictions about the fundamental particles we might see there it seems that there are all facets of some strange symmetrical shape in a higher dimensional space and I think Galileo summed up very nicely the power of mathematics to understand the scientific world around us he wrote the universe cannot be read until we have learned a language and become familiar with the characters in which it is written it is written in mathematical language and the letters are triangles circles and other geometric figures without which means it is humanly impossible to comprehend a single word but it's not just scientists who interested in symmetry artists to love to play around with symmetry they also have a slightly more ambiguous relationship with it here's Thomas Mann talking about symmetry in the magic mountain he has a character describing the snowflake and he says he shuddered at its perfect precision he added deathly the very marrow of death but what I just like to do is to set up expectations of symmetry and then break them and a beautiful example of this I found actually when I visited a colleague of mine in Japan professor Cora Kawa and he took me up to the temples in Nikko and just after this photo was taken we walked up the stairs and the Gateway you see behind has eight columns with beautiful symmetrical designs on sever them them are exactly the same and the eighth one is turned upside down and I said to professor Kurokawa Wow the architects must have been really kicking themselves and they realized that you know they made the mistake and put this one upside down he said no no no it was a very deliberate act and he refer me to this lovely quote from the Japanese essays in idleness from the 14th century in which the SAS in everything uniformity is undesirable leaving something incomplete makes it interesting and gives one the feeling that there is room for growth even when building the imperial palace they always leave one place unfinished but if I had to choose one building in the world to be cast out on the desert island to live the rest of my life being an addict of symmetry I would probably choose the Alhambra in Granada this is a palace celebrating symmetry recently I took my family we do this rather kind of nerdy mathematical trips so which my family love this is my son Tamir you can see he's really enjoying our mathematical trip to the Alhambra but I wanted to try and enrich him I think one of the problems about school mathematics is it's it's it doesn't look at how mathematics is embedded in the world we live in so I wanted to open up his eyes up to how much symmetry is running through the Alhambra and you see it all really immediately you go in the reflective symmetry in the water but it's on the walls where all the exciting things are happening the Moorish artists would deny the possibility to draw things with Souls so they explored a more geometric art and so what is symmetry and the Alhambra somehow asks all of these questions what is symmetry when a two of these walls do they have the same symmetries can we say whether they discovered all of the symmetries in the Alhambra and it was Galois who produced a language to be able to answer some of these questions the Galois symmetry unlike for Thomas Mann which was something still and deadly the Galois symmetry was all about motion what can you do to a symmetrical object move it in some way so it looks the same as before you moved it I like to describe it as the magic trick moves what can you do to something you close your eyes I do something put it back down again and it looks like it did before it started so for example the walls in the Alhambra I can take all of these tiles and fix them at the yellow place rotate them by 90 degrees put them all back down again and they fit perfectly down there and if you open your eyes again you wouldn't know that they've moved but it's the motion that really characterizes the symmetry inside the Alhambra but it's also about producing a language to describe this and the power of mathematics is often to change one thing into another to change geometry into language take you through perhaps push you a little bit mathematically so brace yourselves push you a little bit to understand how this language works which enables us to capture what is symmetry so let's take these two symmetrical objects here let's take the twisted six pointed star fish what can I do to this starfish which makes it look the same well there I rotated it by a sixth of a turn and still it looks like it did before I started I can rotate by a third of a turn or a half a turn and put it back down on its image or 2/3 of a turn and a fifth symmetry I can rotate it by five sixth of a turn and those are things that I can do to the symmetrical object which make it look like it did before I start it now for Galois there was actually a sixth symmetry can anybody think what else I could do to this which would leave it like it did before I started I can't flip it because I put a little twist on it term tie it's got no reflective symmetry but what I could do is just leave it where it is pick it up and put it down again and for Galois this was like the zeroth symmetry actually the invention of this number zero was a very modern concept 7th century AD by the Indians it seems mad to talk about nothing and this is the same idea this is a symmetrical to everything has symmetry where you just leave it where it is so this object has six symmetries and what about the triangle well I can rotate by third of a turn clockwise or a third of a turn anti-clockwise but now this has some reflectional symmetry I can reflect it in the line through X or the line through Y or the line through Z five symmetries and then of course the zero symmetry where I just pick it up and leave it where it is so both of these objects have six symmetries now I'm a great believer that mathematics is not a spectator sport and you have to do some mathematics in order to really understand it so here's a little question for you and I getting a look of a prize at the end of my talk for the person who gets closest to the answer the Rubik's Cube how many symmetries does a Rubik's Cube have how many things can I do to this object and put it down so it still looks like a cube okay so I want you to think about that problem as we go on and count how many symmetries there are and there'll be a prize to the person who gets closest at the end but let's go back down to symmetries that I got for these two objects what Galois I realize it isn't just the individual symmetries but how they interact with each other which really characterizes the symmetry of an object if I do one magic trick move followed by another the combination is a third magic trick move and here we see Galois starting to develop a language to see the substance of the things unseen the sort of abstract idea of the symmetry underlying this physical object for example what do I turn the starfish by a sixth of a turn and then a third of a turn so I given names the capital letters ABCDE F are the names for the rotations so be for example rotates the little yellow dot to the be on the starfish and so on so what if I do B which is a sixth of a turn followed by C which is a third of a turn well let's do that a sixth of a turn followed by a third of a turn the combined effect s is if I just rotated it by half a turn in one go so the little table here records how the algebra of these symmetries work I do one followed by another the answer is its rotation D half a turn what if I did it in the other order would it make any difference well let's see let's do the third of the turn first and then the sixth of a turn of course it doesn't make any difference it still ends up at half a turn and there's some symmetry here in the way the symmetries interact with each other but this is completely different to the symmetries of the triangle let's see what happens if we two two symmetries with a triangle one after the other do a rotation by a third of a turn anti-clockwise and reflect in the line through X well the combined effect is if I just done the reflection in the line through Z to start with now let's do it in a different order let's do the reflection in X first followed by the rotation by a third of a turn anti-clockwise the combined effect the triangle ends up somewhere completely different it's as if it wasn't reflected in the line through Y now it matters what order you do the operations in and this aware nabel's us to distinguish why the symmetries of these objects they both have six symmetries so why shouldn't we say they have the same symmetries but the way the symmetries interact enable us we've now got a language distinguish why these trees are fundamentally different and you could try this when you go down the pub later on take a beer mat and rotate it by third quarter of a turn then flip it and then do it in the other order and the picture will be facing in the opposite direction now Galois produced some laws for how these tables how symmetries interact it's always like little Sudoku tables you don't see any symmetry twice any row or column and using those rules he was able to say that there are in fact only two objects with six symmetries and they'll be the same as the symmetries of the triangle or the symmetries of the six pointed star fish I think this is an amazing development it's almost like the concept of number being developed for symmetry in the frontier I've got one two three people sitting on one two three chairs the people and the chairs are very different but the number the abstract idea of the number is the same and we can see this now we go back to the walls in the Alhambra here are two very different walls very different geometric pictures but using the language of Galois we can understand that the underlying abstract symmetries of these things are actually the same for example let's take this beautiful wall with the triangles did a little twist on them you can rotate them by a sixth of a turn if you ignore the colors we're not matching up the colors but the shapes match up if I rotate by sixth of a turn around the point where all the triangles meet what about the center of a triangle I can rotate my third of a turn around the center of the triangle and everything matches up then there's an interesting place halfway along an age where I can rotate by 180 degrees and all the tiles match up again so rotate along half way along the edge and they all match up now let's move to the very different-looking wall in the Alhambra and we find the same symmetries here and the same interaction so there was a sixth of the turn a third of a turn with as nth pieces meet and then the half a turn is halfway between the six pointed stars and although these walls look very different Galois has produced a language to say that in fact the symmetry is underlying these are exactly the same and it's a symmetry we call six three two here's another example in the Alhambra this is a wall a ceiling and a floor they all look very different but this language allows us to say they are representations of same symmetrical abstract object which we call 4-4-2 nothing to do with football but because of the fact that there are two places where you can rotate by a quarter of a turn and one by half a turn now this part of the language is even more because Galois can say did the Moorish artists discover all of the possible symmetries on the walls in the Alhambra and it turns out they almost did you can prove using Galois language there are actually only 17 different symmetries that you can do in the walls in the Alhambra and they if you try and produce a different wall with its 18th one it will have to have the same symmetries as one of these 17 but these are things that we can see and the power of Galois mathematical language is it also allows us to create symmetrical objects in the unseen world beyond the two-dimensional three-dimensional all the way through to the 4 5 or infinite dimensional space and that's where I work I create mathematical objects symmetrical objects using Galois z-- language in very high dimensional spaces so I think it's a great example of things unseen which the power of mathematical language allows you to create so like Galois I stayed up all last night creating a new mathematical symmetrical object for you and I've got a picture of it here well unfortunate isn't really a picture if I could have my board at the side here great excellent here we are this is unfortunately I can't show you a picture of this symmetrical object but here is the language which describes how the symmetries interact now this new symmetrical object does not have a name yet now people like getting any names on things on sort of craters on the moon or new species of animals so I'm going to give you the chance to get your name on a new symmetrical object which hasn't been named before and this thing species died away and moons kind of get hit by meteors and explode but this mathematical object will live forever it will make you immortal in order to win your win this symmetrical object what you have to do is to answer the question I asked you at the beginning how many symmetries of the Rubik's Cube have ok I'm going to sort you out rather than you all shouting out I want you to count how many digits there are in that number okay if you've got it as a factorial you have to expand the factorial okay now if you want to play I want you to stand up okay if you think you can you've got an estimate for how many digits right we've already got one competitor here yeah you all stay down he wins it automatically okay excellent so we've got four here five six great excellent after that I should get us going all right anybody with five or less digits you've got to sit down because you've underestimated five or less digits so a hundred thousands of thousands you've got to sit down 60 digits or more you've got to sit down you've overestimated 20 digits or less sit down Oh 20 how many digits are there in your number two so you sort of sat down earlier let's have the other ones who said oh they said the other ones who sat sat down during the 20 up again okay if I told you 20 or less stand up because we're this one I think there are a few here you've just said the people who just last sat down okay how many digits do you have in your number ah ha ha how many 21 ok good how many do have a new one 18 so it goes to this lady here 21 is the closest they actually has the number of symmetries in the Rubik's Cube has 25 digits so now I need to name this object so what is your name I need your surname groups the symmetrical objects generally spell it for me G H e Z now so2 s already been used as you in the mathematical language so you can't have that so gets there we go that's your new symmetrical object you are now immortal and if you'd like your own symmetrical object I have a project so raising money for a charity in Guatemala where I will stay up all night and devise an object for you for a donation to this charity to help kids get into education in Guatemala and I think what drives me is a mathematician are those things which are not seen the things that we haven't discovered and it's all the unanswered questions which make mathematics a living subject and I always come back to this quote from the Japanese essays denying idleness in everything uniformity is undesirable leaving something incomplete makes it interesting and gives one the feeling that there is room for growth thank you

Channel: TED-Ed

Views: 34,984

Keywords: \Marcus du Sautoy\, \Oxford, Science, Ambassador\, \sexy, maths\, Symmetry, math, mathematics, TED, TED-Ed, \TED, Ed\, TEDEducation

Id: 1xlAaBrIxMs

Length: 18min 19sec (1099 seconds)

Published: Sat Aug 17 2013