Marcus du Sautoy: Symmetry explained

the early hours of the morning of the 30th of May 1832 a gunshot was heard ringing out across the fields in the 13th arrondissement in Paris a peasant who heard the shot ran towards where the sound came from and on the ground he found a young man writhing in agony his name was évariste Galois a well-known revolutionary at the time he'd obviously been shot by a dueling wound he was taken to the local coaching hospital where he died the next day in the arms of his brother the last words he said were do not cry he pleaded it takes all my courage to die at the age of 20 but it isn't for contributions to revolutionary politics that we remember Galois the young revolutionary had stayed up the whole previous night trying to articulate a new theory a revolutionary theory in mathematics that he developed maybe was the lack of sleep which contributed to his terrible shot that morning but in that package that he left behind was the beginnings of a new language called group theory which would finally help mathematicians to articulate one of the most important concepts in the whole of nature namely symmetry and it's for helping to complete an epic saga written in this language that John Thompson and Jack teats are being rewarded with this year's arbel prize symmetry is a central concept in science and the arts in science it helps us to explain the behavior of molecules viruses crystals is an important indication of good gene structure it's helped us to understand the fundamental particles of nature it's also important in technology as well the codes that are used to encode digital data use symmetry to keep this data in the integrity of this data for artists to symmetry has always been important and underlies many of the different arts from music to poetry from architecture to painting also in the games that we play from the ancient game of or found in Babylon in 2500 BC played with symmetrical tetrahedral dice through to the modern Hungarian Rubik's Cube symmetry underlies many of these games but it's only in the 19th century that we finally had a language to be able to really answer the question what is symmetry the saga begins with an extraordinary revelation contained in Galois z-- manuscripts that just as molecules can be decomposed into atomic atoms things like hydrogen and oxygen just in the same way the numbers can be broken down to indivisible primes Galois realized that symmetry as well symmetrical objects can be broken down into indivisible symmetrical objects so-called simple groups these symmetrical objects are the atoms of the world of symmetry Galois is breakthrough made us realize that mathematicians could produce a periodic table of symmetry which would be as influential as the periodic table has been to the chemists and prime numbers are actually at the heart of the first of the building blocks in this classification of symmetry for example if I take a 15 sided figure the symmetries of this figure or symmetries for a mathematician of the ways that I can pick this object up turn it in some way and place it back down on an image so it looks like the shape hasn't moved at all so for example I could pick up this shape turn it by a 15th of a turn put it down again and the shape looks like it did when I started but the symmetries of this 15 sided figure can be built out of the symmetries of a Pentagon sitting inside it and a triangle these are the building blocks of the symmetries of the 15 sided figure why well because 15 can be broken out down into three times five this is at the heart of why the symmetries of this object can also be built out of the Pentagon and the triangle for example if I want to move the green dot around to the yellow dot by a fifteenth of a turn how can I use the symmetries of a Pentagon will I turn the Pentagon by two-fifths of a turn which takes the green dot all the way around to the blue dot now I use the triangle to make a third of a turn backwards to all the combination of these two dots all the way background to the yellow dots so the combinations of the symmetries of the Pentagon and a triangle can be used to build the symmetries of the 15 sided figure and the reason this is true is because 1 over 15 is equal to 2/5 minus 1/3 so the symmetries of prime sided figures are the first building blocks the first atoms in this periodic table of symmetry and it's one of the achievements that John Thompson is being recognized for for the arbel prize that these building blocks help you to build many of the symmetrical objects in the mathematical world he proved with Walter Fights the late water finds that if an object or a structure has an odd number of symmetries then that symmetry can be broken down into these prime sided symmetries so the symmetries of an object with an odd number of symmetries are built out of these indivisible prime sided figures it was an amazing theorem it first it was a theorem which made us realize we had the possibility actually to start to classify the building blocks of symmetry it was an epic theorem in other ways as well it ran to 255 pages and it took up the whole of the Pacific Journal of mathematics in 1963 possibly was the longest proof at that time ever written in the history of mathematics now although Thompson's work revealed that these prime sided figures are the heart of many of the symmetrical shapes in the mathematical world not all shapes could be broken down into these prime sided shapes take for example the humble football football or classic football is made out of Pentagon's and hexagons now the symmetries of this figure the ways that I can move the shape such of the Pentagon's and hexagons realign in the same way you find that there are 60 rotational symmetries that I can make of the football now 60 is an even number so Thompson and Fights theorem doesn't apply to this shape but maybe there's another way to break this down into symmetries of these prime sided shapes now 60 is a very divisible number it's one of the reasons the Babylonians chose it for their number for their base for their number system so also the reason we have 60 minutes in the hour but despite the fact that 60 is an incredibly divisible number Galois was able to prove that the symmetries of this shape the 60 rotational symmetries are as indivisible as if it was a prime sided shape so this turns out the symmetries of the football turned out to be one of the new indivisible symmetries in this periodic table of symmetry and it turned out just to be the tip of the iceberg but if we need to see some of the other shapes they're on their list on this list we have to move them the three-dimensional world to the world of hyperspace and we have to look at symmetries of objects in higher dimensions and it turns out that the symmetries of hypercubes actually help us to see some of more of these symmetries in the periodic table so what does a mathematician mean when I say a cube in four dimensions how do I see a cube in four dimensions well if any of you have used a map to come here to the Norwegian Academy you've used a language maybe which helps to translate geometry into numbers and this language developed by Descartes and the 17th century produces a wonderful dictionary to change shapes into numbers so for example a stat nav will tell you that the location of the Norwegian Academy here in Oslo you take ten point seven steps east and fifty five point nine steps north and you will arrive here at the Norwegian Academy and so we can change any position on the earth into two numbers so position space being changed into coordinates numbers which identify the place on the earth if I want to identify my space in space I might need three coordinates to identify it so these coordinates help you to change geometry into numbers so for example a square I can translate into numbers by identifying the coordinates at the corners of the square so we have Greenwich down to zero zeros the corner of the square and if I move one step east one step north or one step east and north I will identify the all corners of the square so I've changed the square into four pairs of numbers which identify the square shapes in two numbers three dimensions well now I need three coordinates to keep track of northeast and also up some some up directions so there I can translate the cube into starting at zero zero zero all the way through to the extremal point at 1 1 1 now the shapes run out unfortunately the pictures run out but the numbers don't and this jaunt of this dictionary of des cartes and almost me to actually conjure up a four dimensional cube is described by four numbers starting at zero zero zero zero and the extremal point of the hypercube is at 1 1 1 1 and using this language of numbers I can tell you that a hypercube has 16 corners 32 edges 24 square faces and is built out of 8 cubes and using this language I can explore the symmetries of these objects now we can actually see shadows of these shapes if anyone's been to the arch at la de Faunce in Paris you will seen will have seen a shadow of a 4 dimensional cube just in the same way as the artist to represent a three-dimensional cube on a two-dimensional canvas will draw a square inside a square and join up the edges the architect at glad a force to represent the shadow of a four-dimensional cube has put a three-dimensional cube inside a larger 3-dimensional cube and this is in fact a shadow of this shape but to really explore the symmetries of these objects you really need this language of numbers to be able to do it now the symmetries of these hyper cubes turned out to be the building blocks are some of the new indivisible shapes and these are actually called early groups named after a Norwegian mathematician sophis Li who began his investigation of these groups actually in prison in France after he'd been mistakenly arrested as a spy during the franco-prussian war now the symmetries of these hyper cubes in higher dimensions turned out to be just 1 over 16 new families of early groups and it's for unlocking the secrets of these groups for which the Belgian mathematician Jacques tits is being recognized with the award of the Arbel prize tits constructed geometrical settings things he called buildings in higher dimensions which help explain the symmetries of these new families now the periodic table of symmetries seemed to be shaping up very nicely except there were five strange symmetrical shapes that have been discovered by a French mathematician called Emiel Matti oh that didn't fiends seem to fit into any of these patterns of nice families of groups that have been discovered discovered up to this point they seem to be little like little orphans sitting there without any families one of these symmetrical objects you can in fact listen to it is at the heart of a piece of music written by Messier called Ile de full - and the two themes that he use if you translate them into mathematical symmetries actually generate one of these objects these strange objects which we call a sporadic group it's actually called m12 so here's a symmetrical serenade to celebrate our our bell laureates this morning [Music] so certainly sounds rather sporadic and that is one of the five sporadic groups that mal Maggio came up with but are there any more well in 1965 John Thompson received a letter from a Croatian mathematician called Yanko claiming to discovered a new indivisible symmetry a sixth sporadic group at first John Thompson was a little dismissive of the claim but as he analyzed yonkos proposal he realized the creation creation could be on to something yonkos discovery turned out to be the beginning of a crazy period in the story of symmetry where mathematicians discover the whole range of strange is divisible sporadic groups of symmetry that didn't seem to fit into any of the patterns of previous generations now many of these discoveries depended on a formula developed by John Thompson which helps you to predict whether there would be one of these sporadic groups out there often the birth of these sporadic groups mirrored in some sense the discoveries of fundamental particles often the physics would predict there would be a particle there before you actually observed it and in the same way this formula that Thompson developed was used to say for example there might be a symmetrical object indivisible with six hundred and four thousand eight hundred symmetries before it was actually constructed the construction would then often depend on finding the right geometric setting to be able to realize that number of symmetries and both Thompson and tits are amongst those who have their names attached to some of these new sporadic groups of symmetries that appeared over the decades since the Yankers discovery in 1965 the culmination of this period of investigation was the discovery of a twenty sixth sporadic group called the monster she's like a snowflake which only appears when you move to one hundred and ninety six thousand eight hundred and eighty three dimensional space suddenly this strange object appears whose symmetries have more symmetries than there are atoms in the Sun yet is as indivisible as if it was a prime sided shape and it doesn't seem to fit into any patterns at all but this twenty sixth sporadic group turned out to be the last of these groups and we're now coming to the realization that maybe we have a complete list of all the building blocks of symmetry they're contained in a wonderful thing we call the Atlas a finite simple groups this is our periodic table of symmetry and it's thanks to the work of art no bow our bell laureates this morning John Thompson and Jack tits that we were able to complete this building blocks of symmetry perhaps one of the greatest achievements of the 20th century and now it's up to the next generation mathematicians to see what we can build with these building blocks in this periodic table thank you [Applause]

Channel: The Abel Prize

Views: 277

Rating: 5 out of 5

Keywords: Marcus du Sautoy, John Griggs Thompson, Jacques Tits, Abel, Abel Prize, Math, The Abel Prize, Mathematics, announcement, 2008, algebra, group theory, N-group, symmetry, what is symmetry, simple group, simple groups, prime numbers, prime number, prime sided figure, prime sided symmetry, hyper space, hyper cube, hyper cubes, 4 dimentional cube, lie groups, lie group, higher dimentions, sporadic group, sporadic groups, the monster, finite group, finite groups

Id: ZAPOEgD_arw

Length: 15min 26sec (926 seconds)

Published: Mon Dec 23 2019