What will happen next? Chaos, mathematics and lemmings | Marcus du Sautoy | TEDxWhitehall

Science and Mathematics probably the best tools we have to try and understand the things that we do know but over the last three years I've been on a journey to try and understand how much science and mathematics can tell us about what we do not know this is a subject of a new book documenting that three-year journey I think that actually we spend a lot of time trying to know what's going to happen next probably especially in government you would love to know what's going to happen next the fact that we would like to know what's gonna happen with the climate or maybe with a an asteroid that might hit us or or even the economy and I would say that's actually my subject of mathematics is probably one of our best tools in trying to understand what might happen next not exactly about looking for patterns looking at the data that's happened up to this point understanding that data to be able to make some predictions into the future but how powerful is that tool in actually getting the answers right I guess the ultimate symbol of the kind of unknowability of the future is the casino dice if I throw this on the table how do I know what's gonna happen next that I'm going to get in this case a five well I guess the hero in this story is always at Newton because he gave us the tools to be able to give us some feeling that we might be able to know what's going to happen next the laws of motion the calculus enables us to understand is sort of world in flux so post Newton we really felt like we should be able to know what that dice is going to do it's just following some Newtonian dynamics as it falls through my hand onto the table impacts I should be able to find out what this dice is gonna do next in fact this is a dice that I picked up whilst I was in Vegas I actually went to Vegas to the craps tables to try and use my mathematical abilities to make a lot of money and for sure I lost a lot of money but they did let me keep the casino dice that we played with and and I guess if Newton is my hero then the Nemesis in this story is this guy here French mathematician ornery punk RA who at the turn of the 20th century understood that even if we have the equations that describe the way the universe evolved they may not be able to help us in predicting what's gonna happen next and this was this a discovery of something called chaos theory chaos theory actually is a statement that even simple systems can have very complex dynamics that are very hard to predict now my favorite example of chaos in action is a very simple system indeed it's a pendulum a pendulum is so simple and predictable that we use it to keep track of time but this is a slightly different pendulum it's two metal pieces jointed together a little bit like a leg the geometry is very simple the geometry of these two pieces are metal very easy to describe mathematically the equations of motion which control the pendulum are also quite straightforward but try to predict the behavior of this pendulum what it's going to do next is almost impossible I think the fact that some of you laughing is indicate an indication of the unpredictability of it humor is related to the fact that you really don't know quite what's going to happen next but the real challenge of chaos theory is can i reproduce that result if I start that pendulum off at exactly the same point then Newtonian physics says it will do exactly the same thing now I have a little notch here which I'm going to which I try and do a repeat of that behavior so try and remember what it just did and is it anything like the following well that's kind of a lazy one whoa there's suddenly off it goes Wow I mean certainly didn't do anything like that last time even knowing when it's finished when is it done its last little one this is my favorite desktop toy I can play with this for absolutely hours so if you'll indulge me let's just do it one more time because it is oh look it's off it goes right away that's so there are three different attempts to repeat the behavior but this is so sensitive to very small changes and completely changes what it will do next my other favorite desktop toy that I have on my table is this one here it's again a pendulum this is a pendulum that swings and I've got six magnets which the pendulum is attracted to and actually I use this to make all my decisions in life so each of the magnets has an answer to a question so we've got Definitely Maybe yes ask a friend try again no way so um I'm a big Arsenal fan so let's asking whether I should make a bet on Arsenal winning the Champions League this season so I'm going to let that one just swing off but a little example of what this pendulum is like so here are three magnets I set the pendulum off it's a track to the magnets but try and predict which magnets the pendulum is going to end up at it's almost impossible looks like one of the top one and then sunny right at the last minute it falls on the bottom one here my one is still going trying to make up its mind whether I'm going to make that bet on the Champions League let's see is it going to make up its mind oh it says maybe okay well that's that's hope I suppose but um but this is a little bit like an asteroid flying around three planets the planets are attracting this the glass Droid and trying to predict you know which planet is going to get wiped out and Poincare I actually discovered the sensitivity of these systems when he was trying to understand whether our solar system was stable and so I'm hearing it three computer models of the behavior of this pendulum so I started the pendulum off in the top left hand corner but I just changed very slightly the decimal place of one of the coordinates of where I'm starting its sixth decimal place you can see it completely change the behavior at the beginning they follow fairly similar paths so the first one ended up at the blue magnet so I've colored the magnets yellow blue and red but very small change in the sixth decimal place and the second experiment it ended up at the red magnet the third experiment I changed six decimal places again ended up at the yellow magnet so although they started off quite similar very quickly they diverged and have completely different behaviour and it's this sensitivity to very small changes which means it's very predictable to predict the future this is a computer graphic which helps you to try and work out what's going to happen next with this pendulum if you start the pendulum over a yellow region it means it's going to end up at the yellow magnet so you can see that there's its large yellow blob that's a region over the yellow magnet if I start the pendulum near a yellow magnet I'm pretty guaranteed that it's going to end up at the yellow magnet the other two magnets can't attract it but interestingly opposite that yellow blob there's another region of quite big yellow where it's also predictable because it just swings backwards and forwards and ends up at the yellow magnet the other two can't pull it apart so there are regions where we can know and of course mathematics has been very powerful as a tool to work out what's going to happen next it's why we've been able to land a spaceship the size of a washing machine on the side of a comet but I started that pendulum off in the top left hand corner and this is an area which geometrically we call a fractal a fractal has infinite complexity so as i zoom in on it it never simplifies it never becomes a single color so as i zoom in I think I've got more and more data about my starting position Izu me in the 50th decimal place but even the 51st can change it from red yellow to blue and this is why if I'm in that region it's incredibly difficult to predict what's going to happen next and I think this is a message here it's it's important to know when I can know when I'm in a predictable region but maybe as important to know when I can't make predictions about what's going to happen next to the system this has a name of course you may have heard of it the butterfly effect because these chaotic systems control so many things in the natural world the weather why can't we predict the weather beyond 5 days because it has incredible sensitivity to small changes so the butterfly flapping its wings in Brazil can cause a tornado to suddenly hit London and it's not just the weather the economy might well have dire chaotic systems in it but also population dynamics so here's a little question for you there's there's an animal which it's rumored every four years it throws itself over a cliff in a mass suicide pact but I got to do a little survey see which one you think so a hands up if you think muskrats throw themselves over cliffs every four years any votes some muskrats in the audience there's one right at the back yeah they are right okay any what about voles who thinks voles actually throw themselves over cliffs every four years yes says oh come through more for the bowl fans so okay great how many things lemmings throw themselves over cliffs well the hands are going up in fact Oh oh he put up his hand I guess I better put my hand up on yes we even have this expression behaving like a lemming it's believed you know that they just follow the behavior of other animals and indeed the story is that every four years the lemming population dies out and that the the theory was that they perform some sort of mass suicide pad they just follow each other you may remember that little computer game did you ever play that one I used to love that one you'd set your lemming off when all the others would follow it over the cliff well it seemed like this behavior had been proved by Walt Disney I'm old enough to remember when Walt Disney used to make nature programs and they went out to the Arctic and they discovered on these lemmings behaving very strangely hurting themselves to the side of the cliff and you think well gosh certain death if they go over that cliff smashed into the the water below do they turn back no they start throwing themselves over the cliff in some sort of suicide pact so this footage from Disney seemed to prove this idea that the lemmings kill themselves in this suicide pact every four years there's one here who really doesn't want to go over he's saying no no no go go no go guys and over he goes as well so so it seemed like the theory about the suicide pact had been proved four years ago a few years ago the cameraman who filmed this sequence came clean those lemmings did not want to go over that cliff in fact the film crew had set up a spinning plate on the side of the cliff and there was somebody in the production team putting lemmings on site on the top of this spinning thing and they were being flown over the side of the cliff they didn't want to go over so it seemed like we were back to the drawing board what was it every four years the the populations of lemmings really plummet so if it wasn't the film crew can't be going every four years I'm putting lemmings on this spinning plate and you know wasn't a sort of massive what was killing the lemmings it turned out it was mathematics that was killing the lemmings I think something that many students would sign up for yes incredibly deadly subject by but actually there's a mathematical formula at work which determines the number of lemmings from one season to the next it's actually very simple formula so here's I've simplified it a little bit so we can explore the dynamics of this formula but basically each generation you get to the next generation you get a sort of multiplication factor of that in this case the lemmings double each season but there's not enough food for all the lemmings to survive and so this formula tells you how many lemmings will not survive that season so what you do is you take the current generation you multiply it by the previous generation and in this case we're going to divide by 10 and then that's the number of lemmings that won't survive that season so I want to actually explore the dynamics of this formula with you and since we're on a stage we're going to actually try and act it out so I need two volunteers I'm gonna start with two lemmings so can I have two volunteers to come up and help me demonstrate embody the mathematics of this this is all about getting involved and taking part so I need two volunteers to be much there's one this you'd serve you could come down and then I need another lemming as well I know to get this going we need a woman I'm sorry Elsa don't worry it's purely theoretical so we are we will be sorry I need could I have a female volunteer yes yeah why don't you come up thank you very much I'm going to start picking on people so yeah going up you come it's great okay so this is our first generation of lemming so we have to roamings and the formula says you double up so there'll be four lemmings in the second generation so I actually knew so yes come on down that's right you're you're you're right and the TEDx talk great X so excellent so we've got two but I now need two who are they're gonna be the offspring of these don't worry we're not going to do this other than in the mind all right so can I have two more volunteers who are going to be it's a great excellent annoyingly right in the middle of the thing god that's great so one more volunteer please else I will start picking on people great here excellent right so here are our four lemmings who have grown over the course of two this generation but they're not all going to survive and the formula says I take four multiply it by 2 that's 8 divided by 10 that's roughly 1 is not going to survive so 3 will survive one will not so how we're going to sort out which of these lemmings is gonna die oh look there are some chairs in the wings that's great um why don't we use the chairs and there's kind of like nice circle yeah great that's one so three are gonna survive so I need three chairs you probably know what I'm going to ask you to do next okay so one is not going to survive so we're gonna play musical lemmings and you've got to fight for these chairs okay well we needed a bit of music to get you going so when the music stops you gotta fight for a chair and one who doesn't get a chair gets kicked over the cliff okay so cue the lemming music you know and some lemming dancing kind of helps here although don't call me ya know hovering okay no hovering there sir yeah a little bit of clapping also but which three are going to survive one will not survive oh wow that was quite a fight out there at the fronts I think but I'm afraid you didn't survive so this one gets kicked over the cliff there he goes right and we've got three survivors so let's get our three survivors up okay thank you oops yes let's see so three lemmings doubles up to six so I need three more lemmings for the next round so could I have three more volunteers who thinks they can beat at this lady here - that chair okay we got a volunteer here I need two more of course we've got a dance in front of camera I definitely yes good excellent and one more great you could have played the music for us I think yeah so one more would you like to come up no but you are going to come up thank you very much excellent so so now we've got three lemmings double up to six so remember the formula which tells us how many you're going to survive three times six 18/10 that's roughly two are not going to survive four will so we need a another chair in here let's see whether the wings yes excellent so you've got one more chair so we've got a square here okay so who's gonna survive who's gonna not so unless cue the lemming music as she's two bites the dust' this time so which two are going to bite the dance let's clap them along bit of lemming music living dancing excellent you're hovering I can see it you are hovering so let's see who - don't survive Oh - gosh I sound gentlemanly look if Jessica and this oh my gosh granddad's gone granddad's gone oh dear oh but you're still here grandma's or you kicked your granddad oh my god so we've got four survivors so if you could get up now what happens next will they double up they've got four double up to eight so I need four more volunteers if you want to play again you can you want to get them back so for more volunteers please this is the last your last chance to play excellent there's one coming down there three more please come on this is all about getting involved in bodying the port great excellent yes one there and bring your friend oh no we're too late too late it the place has been taken so now we've got 8 4 times 8 is 32 divided by 10 that's 3 are not going to survive 5 will so we're gonna make another space here I've got one more chair in the wings here so five survivors three I'm not going to sieve alive so let's queue the lemming music so three is the magic number this time three are not going to survive is our grandma gonna make it yes fantastic there are three that went okay over you go let's kick them over the side of the cliff woof woof woof yeah this one's really determined to make it through the whole generation so if you could get up so now something interesting happens to the dynamics now so we've got our five survivors but what happens on the next round five doubles up to ten five times ten fifty divided by ten I have five that died and five that survived so now we get a stability with this formula and although the population started small eventually stabilizes so let's give our surviving lemmings a big round of applause thank you and the dynamics of this formula are such that wherever you start they will always get a fixed generation so but that's not what the lemmings do it the lemmings disappear every four years so what's happening with the lemmings well if I make the lemmings a little more frisky and we triple every generation and we run the formula again we start with two if we ran that game we would go up to five on the first round then to eight and then it will go five eight anyway ping pong between these two values with this new formula and so now we get a periodic behavior it doesn't stabilize if we make the lemmings even more frisky so they're multiplying up by a map factor of three point five um then we see the actual behavior that happens in the Arctic we see four different values one of them which is very low and that's what we're seeing when we see the population of lemmings disappear every four years so this is all very predictable and nice patterns but if I tweak it a little bit more and take their blemings and make them incredibly frisky so they're multiplying by a factor of four then suddenly the patterns disappear and chaos ensues and this is the important thing with these dynamical systems so know the areas where you can make predictions but also to know when actually you lose control in this case if I put one extra lemming in it completely changes the dynamic of the population from one generation to the next this kind of butterfly effect or a lemming effect I guess it would be called um so what about my casino dice I'll return to my casino dice can I know about this actually I talked to one of the people who understood that these population dynamics are chaotic it's Lord may a colleague of mine in Oxford he's actually a cross-party member of the House of Lords he's actually one of the interviewees in my book what we cannot know and I talked to him about - the importance to government of understanding the role of chaotic dynamics in trying to make predictions or not about what's going to happen next and he said to me over lunch in the House of Lords not only in research but in the everyday world of pollak politics and economics we would be better off if more people realized that simple systems do not necessarily possess simple dynamic properties and I asked him how he was getting on sort of trying to persuade the members of parliament and the civil servants about the mathematics of chaos theory and well certainly about the MPs he said they're markers in our intricity mathematics they're mostly interested in their egos here but I returned to my casino dice and I wondered is the reason that I can't predict what's going to happen next with this casino dice because it's chaotic and I got a little bit of a surprise because it depends on where you're playing so there's a piece of research done by some Polish mathematicians they've analyzed how the dice Falls and it depends what table you're playing on so sometimes this dice can be much more predictable than you might think and if you take one thing away from this talk it's the following if you've got a very squidgy table then the pictures which control the dynamics of how the dice will fall are actually not fractal in nature but very clear-cut so yes this little picture here's the top left-hand corner is a picture of the dynamics of the dice as it falls on a very squidgy table as the tights on a table becomes more rigid and it loses less energy then it becomes chaotic and the pictures of fractal in nature I cannot know what's going to happen next but let's take this table here because his are a soft little carpet what you discover is that if you drop a dice it's more than likely to land on the face that is down I know on your hand if the table is actually squiggly squidgy and and soft so this is quite squishing this off so let's see whether I get a 6 in this case I got a 1 and that's why I chose to do mathematics because you really do know what's gonna happen next thank you

Channel: TEDx Talks

Views: 24,646

Rating: 4.858521 out of 5

Keywords: TEDxTalks, English, United Kingdom, Science (hard), Behavior, Control, Future, Games, Humor, Math

Id: g0GOooE_79A

Length: 21min 35sec (1295 seconds)

Published: Mon Oct 17 2016