 I'd like to bring up Robert Cohn from the Courant Institute, a mathematician who will start the evening by delighting you with his own version of mathematics. Thank you very much. It's great to be here. So I'm going to spend five or ten minutes waking you up, I hope. I want to tell you about something that actually came out of my own work, sort of the general type of thing. Are there people who like video games or games of strategy here? Nobody? So this is not video game, but it could be, and it's certainly a game of strategy. It's a two-person game. We're thinking about two players, Paul and Carol, and you see a rectangle there on the screen. So Paul is located where the X is, and Carol is trying to keep him inside the rectangle, and he's trying to get out. And the rules of the game are very simple. He chooses a direction to go in, Carol can't rotate it, but she can reverse it, or she doesn't have to. She could either reverse it or not. And then Paul has to step in the possibly reversed direction, a small step, call it Epsom. So, well, let's see, Paul is towards the top, and you might think he would go up because that gets him towards the boundary, and, of course, that's a terrible idea, because Carol will reverse him and send him down. Any suggestions to Paul? What do you think he might do? Yeah? Go right or left? Go right or left. That's safe in the sense that... It's too close to the edge, no matter what you do. Well, closer to one edge or the other, and anyway, the problem I just described isn't there anymore, right? Okay. But, of course, oh, maybe he got here by going, you know, rect and rotate, or maybe if he keeps on going right or left when he gets here, then what? It's going to be the same problem, right? So, yeah? Go diagonal. Go diagonal. And in fact, okay, so you've got a very important idea, which is that my idea about just looking at this closed as part of the boundary wasn't probably a good idea. So, yes, he needs to do something richer. You can... Okay, we can think about it a little bit better if we say, oh, rectangle is kind of special, how about a circle? From a circle, there's a slightly smaller circle from which he can definitely exit in one step. See on the left? And in fact, so smaller and smaller circles from which he can definitely exit in one to three steps here. And if he had chosen... If Carol reversed him, no problem. In each step, he'll get to the slightly outer circle, right? So for the circle, we understand very well what he should do. If it's an ellipse, well, you can play a similar game. It's just that the things you're going to get by a construction like that one, they're not going to be circles or ellipses. Okay, so by that method, you could find out this curve from which he can exit in one step and two step and so on, yes? I don't understand why she still can't stop him in a circle, just by whatever step he takes. You know, she reverses the direction. If he's where that X is, and of course, it has to be that that line segment is length to epsilon, epsilon to the right and epsilon to the left. If he's one step away, then... He's one... Right. Here, he's two... On this circle, he's two steps away because taking a... He would get in one step up, of course, this line segment has to be length to epsilon. If she reverses him, he gets to the same circle. It's crucial that that line segment is tangent to the circle. Okay, and in fact, very close to that question, thank you for that. Let's... I forgot to tell you Paul is really lazy. He doesn't want to think too much. He wants a very simple strategy, even for the rectangle. And any suggestions? No? He understands circles really well. Yeah? I'm not sure I understand the question. I mean, is Carol standing... Carol is trying to postpone his departure from the region as long as possible. Maybe forever. And the question is, how can he somehow get out anyway? Here's what I'm thinking for a lazy player. So let's pick an origin somewhere in the domain. And if he's on a circle, wherever he is, he's going to be on some circle around that point. And he could just go sideways, that is to say tangent to the circle, his typical distance epsilon. And if he gets reversed, he'll go the other way, but it'll still be distance epsilon. And then, of course, he's a certain distance from the origin. He goes along this line segment, and he's going to be a little farther from the origin, by Pythagoras theorem. Yes? Back and forth? No, he won't go back and forth, because at each step, he'll be on a larger circle. And eventually, he'll be on a circle that lies entirely outside the square. I get it! Okay, good. So by this method, Carol can reverse them or not, but he somehow always makes progress away from the origin. Yes? Can I just clarify something? When you say he, she reverses them, are you saying that he has to take a step in the opposite direction? It's not that she can bring him back to the origin? No, that's right. He has to take a step, and she only controls the direction he moves it. Okay. I'm sorry if I wasn't clear about that. So this is a two-person game. It's a sort of cute two-person game. You might say, oh, what's a mathematician doing, playing games like this? So two-person games arise a lot in economics, recently in my work also in machine learning. There's an area that you could argue whether it's math or applied math or engineering, but it's called operations research. I would call it math. This particular game is sort of special. It's cued exactly because it involves geometry. That step size I called it epsilon because I'm sort of interested in what happens when it's small, and that involves differential equations, and in fact, that's how I got in this business. I studied it in a paper with Sylvia Serfati. So if you're interested in reading something about that, these frames will be on my website. They're very easy to find. Robert Cone Mathematics in Google, you'll find me, and they point you to where that is. So this was, I hope I woke you up. At least I hope I didn't put you to sleep. But the real feature today is Chris Budd. I've known Chris's work since the 1980s, and it's really a pleasure to be able to introduce him today talking about vital math, how mathematicians have changed the world. Thank you. All right. Right. Well, good evening, everyone. So I'm Chris Budd, and I've come all the way over here from England. In fact, this is my very first visit to New York, and I arrived on Saturday nights and spent a fantastic Sunday walking around your amazing city and did, of course, what you have to do, which is I went up the Empire State Building, and I took lots of photos at the top, and I sent them to all my friends and my family just to show them I really am in New York. So it's absolutely a pleasure to be here, and I'm really enjoying your city, and I've also had the opportunity to visit the Courant Institute, which I haven't visited before. And of course, to come here to MoMath, I spent all of today, and I was just overwhelmed not only with all the wonderful exhibits here, but the sheer energy and enjoyment and pleasure of the young kids that were here this morning, and they were just amazingly excited by the whole experience. So I'm from the University of Bath, hands up, who's ever been to Bath in the UK? Loads of you, fantastic. So we really welcome visitors, and it's a city which has buildings 2,000 years old, which is really kind of going some, and you can go to the Roman Bards, and it's still warm, and it's still hot water there, it's just wonderful. My own university is very young, it's celebrating its 50th anniversary this year. I've been there for nearly half of its existence, which is crazy, but I have various other jobs, and one of my other positions is what's called the Gresham Professor of Geometry, which has a distinction of being the oldest professorship in the country, so I am officially the oldest professor in the UK, and this was a professorship that was founded by Queen Elizabeth I, no less, and so it's rather fun to be that as well. So let me just quickly tell you what I do in Bath, so obviously I teach students, but my other big job is to find linkages between what the work that's been going on in the university, the research work in the university, and industry, so industry in the UK or further afield, and trying to find links of mutual benefit between university and industry, so I'm what's called an industrial mathematician, there we are, and what I want to do today is kind of tell you a little bit about my own experiences in working with industry, but also to take you on a sort of historical tour, showing the really important way that maths has contributed to modern civilisation from ancient times to right up to the present day. So I'm a mathematician, I'm reasonably positive about mathematicians, my son's a mathematician, I'm very positive about my son being a mathematician, but on the whole mathematicians don't have a particularly great image, of course this is why we have MoMath to help change that, but we don't have a great image, and to test this I conducted a highly scientific survey amongst my daughter's friends, so I should explain my daughter is doing an art degree in Edinburgh at the moment, and so here are some common views on math and mathematicians, by the way I need to apologise, I might occasionally go into English and call it maths, and I hope you understand, but I'll try and keep with the math. So view number one about math, math is completely useless, lots and lots of people have that view. Here's the other one, another one this is a view my daughter strongly holds, mathematician's evil soulless geeks. So here's a picture of the evil quadratic equation taken from my article 101 uses of a quadratic equation about to devour a poor student, there we are, and here's another one, if you go into Google, which is a wonderful piece of math, I'll tell you a bit about it later in the talk, and type math into, or mathematician into the images, it gives you some interesting pictures, and they all show one thing which everyone seems to believe, all mathematicians are mad, so this is one of the ones I got from Google, there we are, a wonderfully mad mathematician that's even writing backwards, there we are. So these are views, and unfortunately they are views which are quite commonly held, and these are kind of kind of fun views, but some people are really quite frightened of math, or even very suspicious about math, and a somewhat unfortunate example of that occurred, I believe it occurred on a flight from maybe even from New York, where there was a professor, I believe a professor of economics, not even math, but they were, the plane was waiting to take off, so they did what any mathematician would do, whilst you're waiting for a plane to take off, which is take out your notebook and start writing formally, unfortunately this was not a good thing, because the passenger next to them saw their writing formally, and immediately got suspicious, and alerted the cabin crew, and the plane was delayed whilst the person was taken off and interrogated, all for just writing math, okay. So these are problems, and the shame about all of this is not only is it not true, it's really really really not true, that math is basically the basis of the modern world, the modern world would simply not exist without mathematics, okay. The technology that we celebrate today, everything we do is all heavily based on math, in my pocket I have my smartphone, which I've turned off on request, but that is absolutely stuffful of mathematics, and lots of mathematicians work in the smartphone industry. And here's a very, the series is a very good quote, so here's one quote, much of industry, which is who I'm trying to work with, has problems which can be formulated and solved using math, and if you don't believe me, here's Edward David, president of Exxon, ex-president of Exxon, who ought to know what he's talking about, and he says here, too few people recognize that the high technology so celebrated today is essentially a mathematical technology, okay. It's numbers, it's information, it's mathematics. Now there's a problem with all of this, if there wasn't air around me, I'd die very very quickly, okay. If you stuck all the air out of this building, I would die. A math is like that for technology, take the math away, the technology fails, but just like the air around us, it's invisible, and lots of people don't know it's there, okay. And I'm going to demonstrate this with a couple of examples. So I'm going to show a picture of a mathematician, okay. A real mathematician, a mathematician that has changed your life profoundly, more than anyone I could possibly think of, you know. Whilst I'm a great admirer of Washington and Franklin and all that, all these wonderful people, I reckon this guy's changed your life even more than that. So I'm going to show a picture, and all you have to do is shout out from the audience if you think you know who it is, okay. There he is. So any idea who that is? Maxwell, my goodness. This is one of the first audiences I've had, where you've got it right. My goodness, New York's good, you're better than the last audience, actually, in the same building. Well, genius in the room. So are you a math professor or? No, no, okay. Well, I know there are mathematicians in the audience, so I thought I'd better check. Well, absolutely, fantastic. This is Maxwell. But I should say, if I put this picture up in front of a more less knowledgeable audience than you, most audiences haven't a clue who this is. Okay. I tell you, I get various, you know, people suggesting who it is. People, some people say it's Einstein. I say no, he's got a beard. Is it is it Darwin? Sometimes people say and I said no, but the right generation. One person once said is it God? And I say no, it's not God. But the best answer I ever had was from school where I showed this picture and some individual in the school said, is it your mother? And I said, no, no, my mother doesn't have a bit. Anyway, so this is Maxwell. And Maxwell was a Scottish mathematician. He was born in Edinburgh, where my daughter lives. And in fact, you can go and even see where his house is. And then he moved to Cambridge. And in Cambridge, he did some fantastic things. So let me tell you what he did. So the story goes on a little bit more. One of my other other jobs is I'm Professor of Maths at a wonderfully named place, the Royal Institution. Some of you may have seen the Royal Institution Christmas Lectures. Has anyone ever seen those? Maybe they don't get over the, oh, some of that. So Royal Institution was founded 200 years or so ago. And it was one of the first scientific research establishments in the UK. It's in the heart of London. It's by Piccadilly in London. And it's done many very good things. They reckon about 30 elements were discovered in the Royal Institution. A number of Nobel Prizes came out of the Royal Institution. Humphrey Davy worked in the Royal Institution. The Braggs worked there, all sorts of people. But possibly the most famous person to have worked in the Royal Institution was a guy called Michael Faraday, who you may have heard of. And what Michael Faraday is most famous for is he discovered experimentally the link between electricity and magnetism. And he essentially invented the electric motor and the dynamo. And those have then been developed by people like Edison and Tesla into the power generation system we so celebrate today. So very, very important guy Faraday. But he wasn't a mathematician. He knew he wasn't a mathematician. And so he relied on other people to do his math for him. And what Maxwell did was he took Faraday's experimentally derived results. And he turned them into mathematical equations. Now, at this point, I had to issue a slight health warning. This is a math talk. You're here to learn math. So we're going to have some equations. But I generally regard it as useful to warn the audience before I put them up in case you report me and I have to evacuate the building or something like that. So here are equations that Maxwell came up with. There they are. And these are called Maxwell's equations. And if you go to Edinburgh and you see his statue on the base of the statue, you will see these equations. And these equations which link electricity E and B and M, which are magnetism and J is current and row is charge. And these are what are called vector operations. And that's a derivative. So you don't need to worry about those too much. The bottom line is those were his equations. And these are called Maxwell's equations. So what you say they're just equations. What Maxwell then did was that he took those equations and look for solutions. And he discovered that there were solutions which were waves where you have an electric wave and a magnetic wave sort of paired up and traveling together. And he worked out the speed of those waves. And he found that the speed was exactly the speed of light. They just measured that. And he realized that what he'd written down here with the equations for light. And those equations he unified electricity, magnetism and optics on in one set. Well, okay, so he'd written down some things and they knew what light was. But then he did something which mathematicians can do. You can do what if experiments you can say, Well, what if these equations have other solutions? And he found that they had other solutions. And these were waves which had the same speed as light, but a different frequency and wavelength from light. And he discovered these, and we now call them radio waves. So those were the equations for radio. And Maxwell discovered radio by pure mathematics alone. And it was later on that the experimentalist Hertz found them experimentally. And then a bit later on Marconi and others took the theory and turned it into practical means of communication. But it all started with Maxwell. Let's think where we'd be without radio. We wouldn't have radio itself. We wouldn't have Wi-Fi. We wouldn't have TV. We wouldn't have radar. We wouldn't have our mobile phones. We wouldn't have microwave cookers. The world would be utterly different without radio. In fact, the modern world simply would not exist without radio waves. And it was Maxwell that discovered them purely by mathematics. And apart from the splendid audience, which is quite remarkable, very, very few people know who he is. And that's where, this guy should be on the bank notes and stuff. It took us a long time to get his statue up. But quite incredible guy. He's very nice guy, very kind guy. And he did a lot of other things. One of the other things he's very noted for is his work in color photography. He essentially invented color photography. He also wrote poetry as well. He's a good guy. Very good guy. So that's Maxwell. Okay, so that's one mathematician. Probably the math Maxwell is most people don't know who he is. So I'm going to ask another question. Who is the most famous female mathematician? And by the way, everybody in this audience knows who she is. Everybody, without exception. Who's the world's most famous female mathematician? Any ideas? Well, Emmy Noether, excellent mathematician, fantastic mathematician. But if I went into the street, who would know Emmy Noether? Marie Curie, wonderful, wonderful physicist, but not a mathematician, but much more even more famous than Marie Curie. Films have been made about this woman. Ada Lovelace, still famous, but not as famous as this one. I see films, books have been written about her. Hugely famous. Most children would know her name. Any idea? Amelia out. Last time I, wasn't she an airplane pilot? No, I'm going to put her picture up. It's going to surprise you. There she is. Now, who's that Florence Nightingale. Wow. Now, who's heard of Florence Nightingale? Not you. You should read, read different books, sir. Now, Florence Nightingale is an incredibly famous woman, because she, I'll explain, she basically founded Modern Nursing. And she, the story behind Florence Nightingale is that she was sent to Crimea and she set up hospitals in the Crimea, which saved a huge number of lives. And when she went back to England, she developed modern nursing and her practices in modern nursing are used all around the world. And everyone thinks she's a nurse. But actually, she wasn't. She was a statistician. Her, by the way, if you say, oh, I said mathematician in my department in Bath, maths and stats are all in the same department. And so I'm allowed to claim her as a mathematician. She was a statistician. She was one of the first members of the Royal Statistical Society. And it was really good statistician. And the way she cured people wasn't so much through medical care. It's through the kind of more modern approach, which is to try to work out what was causing people to be ill. And so she gathered loads and loads of data on this. And then she produced graphs of this data, which were needed to, to essentially convey what she was doing to politicians, because politicians then and sadly now don't know what numbers are. I think it's true of politicians anywhere in the world. And so she did this through graphical information. And she developed these things called rose diagrams, which are very like pie charts. And so she not only developed medical statistics, which is very, very important. She also developed graphical presentation of data, which is universal. And she's incredibly famous, but no one knows she was a mathematician. Okay. So at least she's famous in England. Maybe she's not so famous in America. I don't know. She she is. That's good. So you do know who she is. Yeah. Okay. And the Royal Statistical Society in England, their building is called the Nightingale Building after her. So there are two mathematicians who have made the modern world possible. One of them, most people don't know who he is. And the other one, everyone knows who she is, but they don't think she's a mathematician. So it's a bit unfortunate, really. However, they have both made the modern world possible. And that shows how important mathematics is in the way the world works. So I just want to show you who uses maths. And the way maths is used in the real world has really changed enormously in essentially my own life. And you heard Robert saying that he, we've been knowing each other for 30 years, and we were comparing notes about what's changed in those 30 years. And the whole thing has changed remarkably, largely because of the development of computers. So 30 years ago, these are the sort of outfits that were typically using mathematics. Telecommunications, that's Maxwell and radio and stuff. The aircraft industry used a lot. Power generation, oil, iron and steel, weather forecasting has been a big user of maths since the 1920s. Security, co-breaking, if you go over that part of this museum, you can see some coding machines. And of course, finance. These are kind of traditional. People think that's what mathematicians do. But nowadays, mathematicians are using maths in a far, far broader capacity. One of the biggest users of mathematics, certainly in the US, is the film industry. The film industry, why do they use math? Because if you have something like Shrek, now I'm really apologised if what I'm going to say is offensive to people, but Shrek actually isn't a real ogre. Sorry if that disillusioned you. What Shrek is, he's not a real ogre. What he is, is a whole load of mathematical formulae inside a computer which some mathematician has come up with and they've constructed those formulae to produce a shape which looks like an ogre. And they've done a great job so that it looks great. So Pixar employs loads of mathematicians. In fact, Tony DeRose came and gave a talk here, didn't he? He's head of research at Pixar, about the use of math in the film industry. He's a massive user of mathematics. One of my other talks I give is about the math if the Lord of the Rings and all the math that was used for that because that was a huge, they use lots of mathematicians there. So film, entertainment, graphic design, the retail industry uses lots of mathematicians. When you go and buy something in a shop, sadly they're collecting data from all of this and they're using that data. If you go on to Amazon and you buy a book it will say people who bought this book also were interested in this and somewhere there's a mathematical algorithm that's doing that. And so the retail industry is using lots, it's great news for young people because it means there's vast numbers of jobs if you stick with the maths. So these are all the sort of outfits that I'm sort of working with in my own job, finding areas where math can be used in the real world. I want to tell you a little bit about the process by which it occurs and that process is then something I'm going to kind of talk a lot more about as I go through this talk. So this is the kind of the process that way maths is used. So this is a bit of a joke. There's a wonderful mathematician called Erdos that said a mathematician was just something that converted coffee into mathematics which is sort of true. But the way it works is this. This museum is full of the most wonderful mathematics, really good stuff. A lot of it is what we would call pure mathematics. And the way you kind of deal with things when I get a problem from industry is you take all the math that you know, all the stuff that you've learned and you try to solve the problems with it. And after a while you find you run out of math. The math that you've learned isn't enough to really solve the problem. So what you have to do is you invent new math. And so you invent new math and then that new math can be kind of abstracted and turned into other stuff and then it can be used to solve new problems. And then you look at those new problems and you find that you need to learn new math from that and so you need to and then that new math tells other newer problems and so on and so on and so on. It sort of cascades with problems generating math, generating problems and so on. And it's a really good virtuous circle and this is kind of how math has developed over many years. Math is also developed by curiosity and just pure abstract reasoning as well. So there's lots of ways it works. So I'm going to take you through this looking at history of the way math has impacted on civilization. So we'll start looking at some of the earliest math and then we're going to do some recreational math and then we'll have a workshop about 20 minutes or so where you have to do something. Okay and it's all going to be fun. It's the stuff on your seats and then we'll go through a few other things and find, I'll try and take you right up to the where we are at the moment and beyond. Okay so that's the plan. Okay so let's start right at the beginning. Early people, where does math come from? Well early people counted on their fingers. Early people counted on their fingers and numbers basically came from that. Why do we know people counted on their fingers? Because we use numbers in base 10 and we have 10 fingers. There's no other reason for choosing 10. 10 isn't a great number for a base. Very few numbers divide into it. If we had 12 fingers we would have been better as it were but we've got 10 fingers and that's why we count in 10s. The Babylonians were a little bit more advanced. They used the knuckles as well and so they counted not only in 10s but in 60s and Babylonian numbers were based around 60 and that's why we have things like 60 seconds in a minute, 60 minutes in an hour and 360 degrees in a circle. That's because of the Babylonians and they have some Babylonian numbers there. So that's the math. Math was kind of invented to count things with. What was it then used for? Well once you've got things like numbers 1, 2, 3, 4, 5 and 6 and you want to start using them you suddenly find that they're not useful for everything. You need to invent some more numbers and so these numbers were expanded to include things like zero which was invented around about the year zero and negative numbers were invented to deal with things like debt and fractions were invented. I mean I suppose you've got three fields and you have five children then each child will inherit three-fifths of a field. So they were invented to deal with that. What was the first application of numbers? Well we're pretty sure that the first application of all of this was something which most people in this audience be very familiar with the taxman. Why can we be sure about this? Because if you go to museums I'm sure there are fantastic places in New York you could go to but in my experience you go to the British Museum in London who's been to the British Museum? Yeah loads of you. They do the math really well there. You can go on a math tour of the British Museum and you can find Babylonian cuneiform tablets with math going back thousands of years and the rhymed papyrus which is an Egyptian thing where there's loads of wonderful math developed and it's all to help the taxman. To answer such momentous questions as I have seven cows the taxman takes five how many do I have now? Okay not enough is the answer of course you know. So yeah and these sort of problems are still important and that sort of math is still used a lot today. But they then discover something else and they had to extend numbers a bit more. Suppose a farmer has a field and that field grows a hundred cabbages. From the taxman the king wants to have a war so they need more taxes out of the farmer and so the taxman says I need 200 cabbages not 100-200. How much bigger should my field be? Twice. Well it needs to have twice the area and the area of a field is proportional to the square of the length of the field. So the question is how much bigger should the length of the field be? And the equation that you have to solve to do that is x squared which is the length equals two. x squared equals two. So this is a very real problem and we know the Babylonians were interested in this problem because here's a cuneiform tablet which I believe is in the British Museum where the tablet is actually trying to solve this equation. These marks here are the various kind of bits describing the equation and even gives the answer. So we know they could do it. The great thing about stone tablets is unlike electronic forms of data they don't go out of date and they last for thousands of years. My floppy disks are useless now. So there we are x squared equals two and they tried to solve this using fractions and they found they couldn't. There was no fraction which equaled the answer to that and so they had to invent another number which is what we call an irrational number to give the answer. So the solution to this is 1.414 21356 237 309 50488 that's to 20 decimal places and it goes on on on on. And these are numbers that they call real numbers and were originally invented for the taxman to work out how to double the area of fields. So as I said the way the way math works at least in my experience is you have these problems you throw the math at them you can they threw fractions at it didn't work they had to go further invent new math and then the math behind these sort of equations became much more important and it was developed and most the problems in the real world like that one we've explained and the great kind of triumph of math around about 1690 was the development of calculus based around these sort of ideas and calculus is now probably the best tool that we have in math to tackle the problems the real world. We had a we had a competition in the UK it was a newspaper competition and the competition was to identify the greatest ever invention the greatest ever invention and and I wrote in and I said the greatest ever invention was calculus. That was my answer it didn't win. If your information the greatest ever invention was apparently the printing press and the second greatest mention was the wheel and the third was fire and they didn't even have calculus on this which is misguided because calculus is out of doubt the best tool that we have to do all of those and much better than all the others but then of course I am biased. Okay so that's all kind of fairly serious maths but a lot of maths doesn't develop by solving problems of practical importance a lot of it come to develops purely out of curiosity or for doing stuff for fun. Who here has done Sudoko? You're doing maths when you do Sudoko and it's good fun isn't it? When you can't do it you know. So I want to kind of have a bit of a diversion now and I'm going to let you loose as well on this to talk a bit about recreational math to make the point that solving puzzles and getting stuck into things and having fun is actually an extremely good way of doing math. Probably the best way in my bit anyway. Now recreational math is a huge huge subject so I thought I'd pick a particular favorite of mine and I want to tell you about mazes and labyrinths which were originally recreational and some of you already been having a go at the things we put on your seats which some examples and we'll have some time to do that in a second don't worry don't worry. One of the earnest examples of recreational maths and a story involving maths is a story about a labyrinth and it involves this chap the Minotaur. Who's heard of the Minotaur? So let me tell you about the Minotaur. The Minotaur was the product of a one-night stand between the queen of King Minos in Crete and a bull. I think the bull was actually his use but he was dressed up as a bull at the time. Turned into a bull or whatever they do. And the product of this one-night stand was the Minotaur who was half man and half bull and he was very very ferocious and he lived in the center of a labyrinth and the labyrinth was underneath the palace of King Minos and there is a picture of it. In a minute I'm going to show you how to draw that okay it's not as hard as it looks. So the story goes as follows Crete had a war with Greece and Greece had lost and as a consequence the Greeks had to send nine young men and nine young women every seven years to Crete and when they got to Crete they were fed into the labyrinth and they couldn't get out and they were then eaten by the Minotaur which is not a great career okay. And they got a bit fed up with this the Greeks so Theseus not unreasonably Theseus who was son of the king of Greece said I will go with the seven young men and nine young men and nine young women and I will attempt to kill the Minotaur. So when he got to Crete he was met by one of my heroes this is a lady called Ariadne. Now we've already discussed Florence Nightingale being a female mathematician. Ariadne in my opinion was the first female mathematician to be recorded in the classical literature or possibly computer scientist one of the two. Why do I think she has this claim to fame? Well she did two good things for Theseus. Firstly she gave him a sword. Oh by the way she fell in love with him but you know these things happen. So she gave him a sword and she also said to Theseus I will give you an algorithm for cracking the labyrinth. An algorithm. Now I'm not going to tell you what it was we'll leave that in two after the break and I'll tell you what the algorithm was. So she gave Theseus an algorithm for cracking the labyrinth and using this algorithm he went into the labyrinth found the Minotaur, killed the Minotaur, got out of the labyrinth and took the young men and young women back to Greece and thus was a great hero. Well he wasn't such a great hero because on his way he stopped off at an island called Naxos where they had a great party and then after that party they sailed off and only after they'd sailed off to they realized they'd left Ariadne behind and she died of a broken heart and turned into a spider or something like that. Anyway hero of the story the hero of the story isn't Theseus no no he left Ariadne behind. As far as I'm concerned the real hero of the story is the labyrinth and I'm going to show you how to draw a labyrinth and when we have the break you'll have a chance to draw one yourself and this design although they found it everywhere in the ancient culture is universal. There are clear examples of Native American populations in the US having essentially discovered the same design and math is universal and the way it works is this in your pack you're going to have one of some help you do this you have a thing called we call the seed which for this design is a cross and four dots like this and four arcs like that so in your pack you have one of these and the way you draw labyrinth so so wait for a minute you'll have a chance to do this in the break you start from the bottom which is this bit here and you draw a little loop to that bit there okay with me so far any confusion there no. Anyway you then take the next one and you join it up to that dot there and then you take that one and you join it up to that line there and then you take this one and you join it up to that one there and then you take this one and you join it up to that one there and then you take this one and you join it up to that dot there and then you take this one and you join it up to that line there and then you take this one and you join it up to that one there there you have the Cretan labyrinth. Isn't that lovely? And that's a design you can find on coins the Romans used it for their mosaics and they put it in cathedrals and as I say it was independently discovered by Native Americans. So that is the Cretan labyrinth and I was part of a group celebrating the 150th anniversary of the London Underground and we put one of these up in different designs on every single underground station in London. If you go around the London Underground you can see these. So that's the Cretan labyrinth. Yes sir. The hearts in this one would be here. So that's where the Minotaur lives. There he is. It looks like Shrek. Yeah he's pretty ferocious isn't he? And here's Thesias. There we are with his sword. There we are. And he's going to go in and find the Minotaur. So you'll have a chance to draw this in a second but before we do that I'll just take you a little bit on. So this is a labyrinth and later on in the 18th century the idea behind the labyrinth was evolved into what we call the modern maze and people who had large houses used to build mazes in their large houses and these mazes were designed to trap the unwary. You'd go into them and walk around in them and find that you'd occasionally get lost in them and people would try to puzzle how to get from the entrance into the centre and that was kind of fun and then Euler who's a great great mathematician designed worked out the math of how you could get into the centre and back again and that math led him into the theory of what is called networks and that all came out of mazes what do we use networks for now? Well the biggest network in the world is the internet. Huge network and Euler's work on mazes is directly used to help us do the internet and what's the thing that uses that Google and that's Google it also uses the theory of mazes developed by this guy called Cayley so the algorithms behind Google go back to the work of Euler studying mazes for recreation. So I'm going to let you have some fun now in your packs you have some mazes I'd like you to have a go at cracking them by the way just to say it's recreation to have we can have some fun doing this and then afterwards I'm going to carry on talking so there will be some talk after this as well but but we'll have a quick break while you try and crack some of the mazes and draw some labyrinths. I want to tell you a little bit about some of the algorithms that have been used to solve mazes the the as I say Ariadne in my opinion was a mathematician and a computer scientist because she came up with this algorithm for solving the maze or the labyrinth and her algorithm was beautifully simple but remarkably effective she gave theses a a ball of of thread and say you tie it to some object outside the labyrinth and then as you go into the labyrinth you unwind it and then whenever you need to get out of the labyrinth you just wind it up again and you'll get out again okay that sounds incredibly simple work very well and that is the basis of a modern computer algorithm called the flood algorithm for solving the labyrinth yes sir no I think it was just an ordinary yarn but but you may be right I need to may need to check on that and now another algorithm is if you ever read the book three men in a boat in that book they go and try and solve Hampton Court maze which is what another one of the ones that you've got and in Hampton Court the book they in the book they say well the way to solve it and Harris is the leader of the gang says you solve it by always turning left or another way is you put your left hand on the hedge and keep it there and that will actually work that will solve Hampton Court maze and it will solve a lot of maze you won't solve all of them but is actually a very good algorithm to try it will always get you out of a maze even even if it won't get you into the center so always turning left is a good algorithm but I will show you that these years as well as being a rotter needn't have had to it was actually very stupid because if we have these years here there he is with his swords and his thread and we got the minor tour here whoops there is if we enter the labyrinth we go like this and we are not really having to make many choices in life at the moment still no choices still not many choices no more choices still no choices we seem to be doing well and and okay so it was probably very bad for that and extremely bad for the minor tour so and a labyrinth with this design is what is called unicursal which means that you can go in and out without actually having to make any decisions at all so we didn't need the thread after all but the other maze is like Hampton Court and Hatfield House and all the other ones you have to do a little bit more so a labyrinth is something where you don't have to make decisions and if you go to London underground you'll see 270 of these on the walls of the underground stations so I will move that away and we will carry on so another area of recreational math or some people take it a lot more seriously is math and music and people often say there's a close link between math and music and mathematicians and musicians and that's absolutely true music uses a lot of math so some musical notes sound better when you play them together there's an early instrument the octave C to C is a good chord C to G is a very good chord universally regarded as good and C to E these are notes which sound good when you play them together the reason all of my stuff is in the key of C is that my musical career never extended beyond scales which didn't have sharp some flats okay so that's why I know this now why is this why do these notes sound good well the reason was discovered by another wonderful mathematician a mathematician again most people will know of but you may not realize he did music anyone know who worked out the maths of music Pythagoras very good so here's Pythagoras he is very very famous for Pythagoras's theorem which was invented by the Chinese about a thousand years before him maybe he rediscovered it but anyway it's got his name on it so who we complain but he absolutely did do the work on on musical notes and what he did was he measured the length of strings of instruments and he compared the lengths with the notes which sounded good together and he came up with an idea of fantastic in brilliance was that he realized that if you do the octave the octave compare corresponds to two strings one being twice the length of the other the note C to G is two strings which from proportion three to two and C to E proportion five to four and Pythagoras found an incredible link between musical harmony and fractions now we understand this now better in terms of the way waves work and the way our brain work but at that time it was incredibly powerful link to make and Pythagoras did something else again this is what mathematicians can do he took the idea further he said well let's suppose that we have a sequence of notes with simple fractional relationships let's see if we can do this so he came up with notes with these relationships which are all simple fractions and that has a very important name we call that the scale so Pythagoras invented the scale so that's the basis of modern Western music goes back to Pythagoras and his invention of the scale and that I hope puts paid to this claim that mathematicians evil soulless geeks okay we we mathematicians very important role in development of music and there's an interesting twist to this in that this scale was used a great deal up to about the 18th century and then in the 18th century stringed instruments were invented and so keyboard instruments invented and with a keyboard instrument you tune it really well but then you start with the tuning with a violin you can vary the tuning every time you play it by moving the fingers and it was found that this scale work brilliantly for one key but terribly in another key and it was because the the the ratios differ from one note to the next and so they the mathematicians developed a different scale where the notes were constantly varying in frequency and the the ratio they needed 12 notes in the octave so the ratio is this wonderful number 1.059 which is the number which were multiplied by itself 12 times gives 2 okay we did the square root to earlier and this is a very important number in in musical harmony and this led to the thing called the well-tempered scale and Bach was very familiar with this work that going on and he so liked the well-tempered scale that he wrote a piece of music called the well-tempered clavier where you go through every key on the harpsichord and they all sound good and it's all based on maths and it's the maths of geometric progressions. Yes, that's correct yes all the way through it's not wonderful it's a geometric progression and it works really really well and it all goes back to his mathematicians working with musicians again we're not evil soulless people at all okay so music changed a lot in the 18th century and something else was changing at the same time which also involved a lot of math so here I am in America I've come here from England I flew okay but my grandfather used to come to America a great deal he was an engineer and he went on the Queen Mary he went by boat okay and of course in the 18th century that's how people traveled they traveled by boat but the problem that they were having was that it was they were finding it very difficult to know where they were so one of the big problems the 18th century was finding your position at sea so here's the earth there's England there's New York I suppose about here you're about here aren't it yeah and we have latitude which is essentially the angle from the equator and longitude which is the angle measured round from of course London yes I've stood on the zero longitude line in Greenwich so what was the problem well the problem was to find your latitude how far up you were and your longitude how far along you were now the first thing they cracked was latitude so you could work out your latitude basically by using this thing which is a sextant which measures angles very accurately and they realized if you measured the angle between the Sun and horizon or between the pole star and the horizon you could actually very accurately find your latitude well you had to use a ton of trigonometry in other words more math in fact trigonometry on the surface of the sphere which is called spherical trigonometry which is a little bit harder but still great fun and trigonometry and the sextant combined together meant that navigators could work out their latitude they could work out how far above the equator they were so I'm going to get to America was that you would sail from England on the same latitude and and when you got to something big that was probably America or possibly Canada I don't know am I allowed to mention Canada here by the way if in case you wonder about the Canadian flag it's because my mother is a whole her family is Canadian there we are sir longitude was the big problem though to find the way round and many mathematicians try to solve it using math alone but the solution was a beautiful one and it's the sort of math I do what I mean by that is the sort of math I do is a combination doing sums on paper but also a lot of work with a computer so it's combining technology with formally and the solution to this problem was due to a guy called Harrison who was a clockmaker and it was based on the observation that the earth goes once around every 24 hours and so if you can time when the noonday Sun is and measure that time on a clock which is the same as a clock in Greenwich for example then by measuring that time you can actually work out how far around the earth you are you have to do a whole ton of other stuff as well but that's the basic idea and that was math but it was also linked up with technology because you needed the clock as well and here's Harrison's clock called H4 which was the first clock which was accurate enough to make that possible and I love this it's a lovely combination of technology and math working together and that is essentially being one of the big drivers of the way things that we're going I say a ton of math was involved as well you have to produce tables of the motions of the Sun and the stars as you appear they appear moving in the sky these had the lovely word of Ephemerides there we are and these were calculated by computers now what's a computer a computer in those days was the room full of people who computed those were computers they were typically youngish people often students and it was found interesting enough that women were better at it than men okay so lots of young women were involved in the calculations and the management on the boats doing navigations had to do long calculations based on these to find where they were but those calculations changed the world why is America such a powerful nation because you could reliably sail your ships from you to Europe and sell all your stuff and that reliability came from navigation and that navigation relied on math now there's an interesting extra story to this the calculations were tedious they took a long time there's also very error prone and so people thought well can we automate these using machines and so a guy called Babbage working with Ada Lovelace who we've heard mentioned who was a very very fine woman computer programmer for essentially the first-ever computer programmer worked together to design machines which would automatically calculate these tables sadly they never managed to build them properly due to engineering problems but Babbage's machine has been recreated this is at Science Museum you turn the handle all the cog wheels go around and it calculated the tables for you brilliant and Babbage's ideas led directly on to the invention of the modern computer with Turing von Neumann and people like that and that all came directly by navigation and the need to do that now I tell you a little bit about my family I am one of four children my sister is a doctor and when we were comparing notes she's really quite similar to me in age in fact we went to university at the same time she said well you're a mathematician you're going to be useless sitting in office doing rubbish I'm going to be saving people's lives okay so she rubbed it in a bit she's like that my sister anyway so we had a little bit of discussion about that but I would argue that mathematicians save hundreds if not thousands if not millions of lives every almost every day who here has been in a medical scanner okay quite a lot of you medical scanners have revolutionised medicine because you can be scanned and you can find out what's wrong with you without cutting you open that's really important and the medical scanner was basically invented by a mathematician a mathematician called Radon and here he is now Radon is one of my favorite mathematicians for two reasons one he did brilliant maths which is used in medical imaging and saves thousands and millions of lives the other is if you ask a school child to draw what they think a mathematician looks like that's probably what he looks like I gave a talk recently in a school and their math teacher was sitting in the audience and he looked exactly like that okay he's got a lot he's bold he's got glasses he's got a moustache in his middle age perfect anyway a fantastic mathematician and he was studying in 1917 a kind of abstract problem he was looking at shadows you know objects cast shadows and he wanted to know if you know the shadows were can you find out what the object was that cast them there's a little demonstration of that over there in the museum just there if you want to interact with that on another day and he wrote down a formula for taking an object and the shadows it cast and then with a bit of genius he also worked out another formula saying you know what the shadows are this is what the object is now I always give you warning as I say before I put formulae up so here's Radon's formula for the shadow cast by an object yay there we are asked for Radon by the way F is the object that's the shadow and here's Radon's brilliant formula for turning a shadow into an object wow okay so what you say well listen listen it gets it gets better it's better so here's Radon's formula now you'll notice 1917 various things happened in 1917 we had the Russian Revolution the Great War was going on in Europe and America came in on the side of the Allies in 1917 and so a lot of people were being injured and they had x-rays then Marie Curie which someone mentioned was driving around in an ambulance x-raying people fantastic x-rays are not very great you could see some detail but not very much and it was realized that Radon's formula here if it could be made to work could turn an x-ray into a really good image that could show you what was going on now the problem with that was they didn't have the technology to do that you had to wait 50 years for computers to be powerful enough to use that and a company called EMI with a guy called Cormac developed the computers developed the scanner to do it and that was the first ever scanning device and there we have it that's the scanner that relies totally on Radon's formula with a lot of other stuff Cormac got the Nobel Prize quite rightly for doing that so that's all based on math these are sort of pictures you get this is someone's head before then it was impossible to x-ray someone's head and see that's the detail that's an image the other way there is a tumor so that image was would allow a doctor to detect and then operate and remove the tumor so medical imaging has utterly transformed medicine hugely reliant on maths in fact it's an error I work in myself you can use it for other things here's a rather elderly person this is toon Carmen approximately 3,000 years old looking good on it and they scanned him to try and work out what had killed him they still don't quite sure various theories scanners are used to clear land mines and some of the work we do at Bath we scan beehives to monitor the bees and the beehives to understand better their behavior patterns and how they react to things like climate changing and so on so scanning is very very important again who's done sudoku who's done killer sudoku killer sudoku well it's a hard type of sudoku but but sudoku and killer sudoku are basically the same maths as scanning so if you ever do killer sudoku using the maths of scanning oh well maybe maybe I only asked Brits to such cruel things as killer sudoku did you do gridler over here no you probably call it something else I should have done my research better I do apologize okay I'll get onto my final case study before we finish and to bring us right up to date so I want to show you one of my favorite photographs ever which you may not think is very wonderful but I think is wonderful and change my life this photo and here it is there okay so that is Saturn okay and you might think well I've seen lots of photos of Saturn so what this was the first photo of Saturn that was taken by a satellite and it was taken in the early 70s by the Voyager satellite why did this change my life because I go and visit my grandparents my grandparents got the National Geographic my grandfather said look at this and this was a photo in the National Geographic the first one from outer space of Saturn unbelievable the technology that that was that was taken by a satellite and beam to Earth using a transmitter with 30 watts of power now 30 watts is less than that light bulb I'm not sure that would probably about 100 watts or maybe more that light bulb where you can see it but imagine we were down in downtown would you see it there possibly but New Jersey would you see it there would you see in California okay which is where they had to get those images to and that was transmitted with 30 watts from Saturn which was millions of miles away all the way the way it was done was they took this picture turned into numbers and then they turn those numbers into a code and that code had lots of information redundant put into it so it didn't matter how far it went and how much it was distorted you could reassemble it at the other end into the picture and then that picture was Saturn I think that's absolutely wonderful it's a combination of math and technology and brilliance and that's so fantastic this technology is used all the time now if you have a CD player then you're using that technology because CD players on the CD you put lots of extra information to make sure when you play the CD it doesn't get distorted or destroyed now here's one of my CDs what you can't quite see and sorry the pictures isn't good are the bite marks the bite marks due to Benji the dog there we are who ate my CD sadly Benji the dog died last year of extreme old age but Monty the dog still around if you want to check my website and he ate my CD and I was most upset because it was my favorite ones when I put it in the machine it still played despite the bite marks and it played because the information on it had all this extra maths which meant even with the bite it could still work where's that math used it's used in CDs mobile phone satellites and so on there they are oh that was sorry actually many more numbers don't worry yes I should change that yes well spotted glass someone's awake okay so this technology is used all over here and it was invented by a marvelous American mathematician working at Bell Labs there he is that's hamming one of my great heroes fantastic mathematician brilliant inventor of what we call error correcting codes which are used to send all the information around essentially created the mobile phone in the internet technology which can send information around but despite that he had a terrible terrible taste in jackets there we are yes it's something like that yeah well that would be done mostly by template matching by comparing it what you what it thinks but but it's things like that you put in extra information so that even if you make a mistake it can still think actually it should be that I mean Google's also using what we call contextual information so it's using the fact that it's expecting people to type in English language but but that's you're getting the idea very much that that we can kind of sort our errors when they arise and that's that's what this technology is and it goes back to hamming but also it goes back even further it goes back to a branch of maths called Galois theory which was invented at the beginning of the 19th century by a guy aged 19 and Galois stuff is heavily used in modern technology now so this is error correcting this things as right up to date with the information age sending huge amounts of information around the world sending stuff around with Google absolutely fantastic so I will conclude as follows so I've basically just in this talk give you a very very very sketchy overview of all the many things that math has done for us the very profound way that math and mathematicians mathematicians have changed the world there's lots and lots more applications out there what is often called pure math which is math developed for interests only suddenly finds applications in remarkable ways and leads to whole new technologies Google was invented by two mathematicians you know who are now very wealthy that'd be a lesson to us all so and you've probably got a sense of the excitement I have being vaguely involved in this and I think they're incredibly exciting times ahead and the great thing is that these are going to be young people driving us forward and I can't wait to see where we get to next thank you very much all right so if anybody haven't any questions we have some time you can raise your hand and I will bring the microphone to you and then if you could stand up can you tell us something about the mathematics involved in making the Harrison timepiece accurate enough the Harrison do I get the name wrong all the timepiece accurate yeah well a lot of it wasn't mathematics a lot it was very very careful bearings but obviously Harrison had to get the the the timing the cogwheels very very carefully balanced to do that but a lot of it was getting the bearings really really good so that they didn't weren't affected by temperature in particular that was the big problem they were having just for the prior questioner there's a wonderful movie called longitude yes that explains the whole yeah and the book by so bell that goes with it yeah thank you for your talk there's a lot of buzz these days about blockchain and Bitcoin and new technologies related to that can you discuss some of the mathematics behind that innovation so that matter finance and stuff like that well that's a essentially a huge extra topic of the way math is involved with finance I can't really touch any and I've essentially avoided finance because I don't know very much about it but I mean one of the really big things in math now is what we call the math of big data which is all related to that as well and that's sort of an extension of this stuff I got on to at the end the math of information hi hi what's your favorite number 31 why ah well all sorts of reasons what's the next number in the following sequence 1 2 4 8 16 the answer is of course 31 for reasons I can explain there's also 31 is the largest number you can count to with the fingers of one hand there are other reasons I like 31 is there a way to have your talk available in every high school in the New York area I need my students to hear especially my average students I believe that these gentlemen here are you are copying it and that you'll be able to get it from the website presume of the memeth and if you were to email me Chris Bud at Bath I can send you the notes and stuff everything so yeah Chris Bud just just look at me up on Google Chris Bud at Bath and and then I can send you stuff yeah all right so one more hand for Chris