Curator as, ada as technical books. And this is Eugenia Cheng. This is made support from the Microsoft Corporation and with media sponsorship by kuw. First, Eugenia Cheng will speak for 35, to 45 minutes and move into the time for a question. If you have a question, please use the microphones on either side of the stage and make sure that your questions are concise and in question form. After the questions, eugenia will sign copies of her latest book beyond infinity, the outer limits of mathematics and books will be for sale and come back, april is packed. Seattle biology panel on the future of health. Paul hawken, ways to reverse warming. And theres so much to go on and you can find our programming at town hall seattle. Org and finally id like to thank the members and our audience tonight. Your contributions allow us to dynamic programming like this and its accessible to the city. Members, this is your town hall. Now its my pleasure to introduce to you our speaker this evening. Eugene cheng is a scientist in residence at the school of the Art Institute of chicago and an honorary fellow of the university of sheffield. Her popular youtube videos use math to explain everything from the perfect glass of wine to the best way to share cake. And has been viewed over a million times. And her latest video over 8 million times and she joined Stephen Colbert on the late show to explore the mathematical wonders of puff pastry. Please join me and give me a warm town hall welcome to Eugenia Cheng. [applaus [applause]. Thank you very much. And thank you for that wonderful introduction. Thank you to the town hall seattle for inviting me back to talk about this book. I had a wonderful time here when i came to talk about my previous book, a couple of years ago. So, thank you all for joining us to talk about infinity and what lies beyond infinity. I was once called in to arbitrate a dispute between my fouryearold nephew and his best friend about whether infinity was a number or not. His best friend that infinity is not a number because my daddy says so and my daddy is a scientist so he knows everything. My nephew sensibly focused on proving that his daddy didnt know everything by reeling off batman facts that his daddy didnt know. I love the fact that infinity is something that children ask about and adults cant answer and for me thats a wonderful place to explore what goes on in the world. What does on in our minds. And to explore and nuture cues 0s cues curiosity in children. And its amazing scientists took thousands of years to talk about infinity that would make sense when you scrutinize it with logic. Unfortunately there are many argument in the world that do not make sense, even when you dont scrutinize them with logic. But certainly when you scrutinize with logic and the whole point of math is to construct arguments that hold up under logic. So thats what were going to look at today. There are infinitely manile things i could say about infini infinity. Loops, paradoxes, infinitely big and what infinity is and talk about dimensions and finish bytalking about about the infinitely small. Loops are one of my favorite things, how i think i first encountered infinity. I had a group with a chimney in the middle and a fireplace and the yoo i had is all the other rooms went around the fireplace and heated by the fireplace. And my sister and i chased each other and because they were in loops, we could chase other around the house. We could change directions and have hysterics and crash on the other side. And loops are the way thats infinity and my alltime Favorite Computer program is one i wrote when i was five. My mother taught me to Program Computers when i was five and i ner got better, and i still program like a fiveyearold. I was so entranced by my favorite program, i never got bored of it. Im not easily bored, which has down sides which means i never got better programming. I could do the same program over and over again, programmed in basic, kind of dates me. 10 hello, 20, go to ten then hit run and you get this, and i could do this over and over again and i did do this over and over again because now we have an infinite loop and this is i think probably my first, along with chasing my sister around my house was my first appearance of infinity, infinity in a loop. And what is infinity . Well, my friend is a basque translator, in basque infinity is the same as 11. Beyond ten it might as well be infinity. My friend sally counting, doi doing and posted on facebook, she had three dogs since, 10 from 2014 and lots from 2010. 5. And somehow beyond 10 is lots. And higher means anything higher than three. So in higher dimensional is goes 1, 2, 3, and after 3 you might as well be doing infinitiment so, it depends what you like about numbers and what situation youre in, what counts as big. Another way we can get a loop is if we sing a round. Id like to try to see if we can sing it together. Everyone here first, here second, here third, fourth. Row, row read . Row, row, row your boat row, row, row your boat row, row, row, row your boat mercedes merrily, mer ily, drive is, but a dream. Guest hey, we going on forever. There was one incident with my nephew again, my sister and i were walking him home and the only thing that stopped him screaming if we sang row, row, row your boat in a round and didnt stop. And here is moo i my depiction of what we just did. And using row, row, row your boat, the next people start here, the next people start here, next people start here. And im happy you spontaneously carried on and when you got here you started again and the yellow people started again and the green people started again. Oops. And the orange people, also started again. And then we just kind of kept going and made the loop. If you look at the four lines in each section theyre the same four, they just rotate around the four groups of people. So this loop is the infinite loop of row, row, row your boat. And we could keep going on forever because its a loop and go around as many times as we like. So, this is another way we get to access infinity. And lets talk about some paradoxes now. If we think about what infinity is and were not careful about it, we run into some issues. And the issues there are two things you can do when you run into problems, either run away from it or go hmmm something looks interesting here and ill investigate why the paradox is happening. If youre a mathematician, you investigate why the paradox is there rather than running away from it. Here is one i really like, Hilbert Hotel. Imagine in your head you have an Infinite Hotel. So your Infinite Hotel has rooms 1, 2, 3, 4, 5, 6, 7, 8, on forever. This isnt possible in real life, but in math its not real life. It means you can do things actually not possible in real life. In relife there are restrictions, time, money, physical stuff. In math the only restriction is logic. So you can do things like have an Infinite Hotel and see what happens. You can have a full hotel and see what happens when one new guest arrives and then you go, well, how about i move the person in room two into room one into room two and i move the person into room two into three. Three to four and so on. And they never run out of rooms to move into because it goes on forever. And that means that room one is now empty and i can put the extra person in room one. This is something you cant do in a finite hotel, but it would have changed the story. Anyway, hilberts paradox was from a couple hundred years ago. This one is more than 2,000 years. They are about motion and how we understand motion. One says achilles can never beat the tortoise. And achilles is fast. And the tortoise starts at the last line here by the time achilles has gone to the line. The tortoise is there, and by the time achilles is there the tortoise is further, that means the tortoise is always ahead of achilles, hes never going to beat the tortoise. That doesnt seem right because probably the tortoise is going to be beaten by achilles, a wonderful video on youtube you might have seen where theres an actual race with a bunny against the tortoise and its perfect because the bunny is going, and the tortoise, and the bunny is, oh, oh, whats over there and everyone is screamy at the bunny to go that and it never does and the tortoise went. I love it. The next paradox says you cant eat all of our chocolate cake because you eat half of it and then you have to eat half of whats left and half of whats left and half of whats left and so on, and there will always be half of whats left left. So youll never finish your chocolate cake and yet i do finish my chocolate cake. And the last one is that nothing is moving. Because if you take a photo if we all wave around and someone takes a photo. In the photo no one will be moving because it only captures one instant and no one is moving, so how is it that were moving. These are things where some form of logic has violated what we understand about the world and some people say that means that math is dumb and other people say oh, we should explore math further to see how we can resolve this using logic. Ill get to that later. The solution of all of this came 2000 years later and thats what calculus is all about. And you might think that calculus is invented to torture people in schoolment and a lot of people said i was okay with math until calculus. And some people said im okay with math until times tables. I have sympathy with both things. Calculus is there to deal with infinity and the infinitesimal small things. Although you think that infinity is far away from the world, considering infinity and how people think about it led to the development of calculus bye controls everything that moves continuously in the world around us. So, everything that uses electricity, everything in the room, anything thats got a curve in it, the curve of the pine piano, the way the strings work. Without that understanding of infinity, we wouldnt have technological developments. Thats kind of a use for infinity. Im a bit wary talking how useful math is because i dont like to think that thats its big selling point. Like, it would be like saying, id like you to meet any friends, theyre very useful. [laughter] so, math is useful, but often what motivates us to do it is the fact that its fascinating, not the fact that its useful. Its great that its useful as well. Here is a picture of me eating cookies. I made some Chocolate Chip Cookie dough and the first is the 1 10 of the whole dough and the next one is 1 10 of whats left and the last ones, ive eaten, they are he going into my mouth. Ill talk about infinitely big things in a little bit. What is infinity . Is it a number . Lets think about big numbers, what does a million looks like. Im thinking about that, ive never seen a million of anything, but theres a great Childrens Book called one million no, how many jelly beansment i dont want to spoil it, but dont want to do a click and the last page is a picture of a million jelly beans and its cool and unfold it because its so big. Here is 10, here is 100. Here is 10,000 dots. I couldnt make this because my computer didnt have enough memory, a pathetic one. Its bigger than numbers, and more than how many people on earth or cells or molecules or atoms. We dont have an infinite million number of things on i think. It doesnt tell us what infinity is. And its like telling yao ming is taller than everyone you know, it doesnt tell you who yao ming is. What is infinity . I said there arent infinitely many things. What about light symmetry on a sir la. I was helping first graders with symmetry and we did a triangle and a square and a h hexaegon. And an ox adgon. It has eight side because an octopus has eight arms. And one said theres hundreds of them. One says millions of them. And the other side you could spend your whole life driving them and you wouldnt finish and the other one colored in the whole circle and said i went home going, what just happened . And i dont understand what just happened. But there are infinitely many lines of symmetry on a circle. So we have infinitely many things as long as theyre not kind of physical things. So, infinity is also a way of winning arguments because someone might say im right times 10. Im right times 100, im right times 1 million, and the other goes im right times infinity. And the other might say im right two times infinity. What if you think that Infinity Plus one, we could say subtract infinity from both sides of the equation, one equals zero, which is not great. We could go Infinity Plus two, and if we keep going, everything would equal zero and even suggest that two infinity is Infinity Plus infinity, is that also infinity or subtract infinity from both sides of that infinity equals zero. Now, everything equals zero. Now were in the zero world, thats okay. Thats a perfectly valid mathematical world just not the one we live in. You cant do very much inside it. Its like the world of candy when i was little because i was allergic to food coloring, i couldnt have candy. No matter how much candy anyone gave me, i had zero candy. A sad world to live in. Infinity isnt an ordinary number you cant manipulate it well, you can, in math you can do anything. If you have a contradiction, you end up in the zero world. You can say that infinity is an ordinary number, but what is it . Its not the real number. The real numbers are the ones that fit on the lines. Here are the whole numbers and in between them, there are all the real numbers, but maybe theres another kind of thing that is. Mathematicians invented this and its called an ordni acordn ordn number and we can do infinity like that, infinity is how many numbers there are. We have to be a little bit careful about that, its how many whole numbers that are and the reason is that this is only the first infinity because there are more infinities beyond that because there are bigger sets than all the whole numbers. What on earth does that possibly mean . Well, lets think about this. The first infinity how many guests fit in the Hilbert Hotel . We could fit a new guest like that, and they fit in the hotel and seems to be telling Infinity Plus one is the same because we can fit that in the hotel. Depends if you care about the hassle that you caused many people. Imagine you checked into a hotel, excuse me, you need to move rooms. Its about the hassle. If you care about hassle. One plus infinity is not the same as Infinity Plus one. That means one guest arrived and then the infinity guest. And you put them in the room. Infinity plus one. You put in infinity guest in first and a new guest arrives and at which point, you have to move them. If youre doing that, Infinity Plus one is bigger than infinity, and this is what i think shakespeare new because in the taming of the shrew, he said if this be not what we look for i have no more to say, but bid farewell forever and a day. Which he evidently thought forever and a day. Forever and a day is longer because you have to wait for forever and then another day after that. So theres a bigger infinity than the whole numbers. Theres a bigger infinity than the number of people in the Hilbert Hotel and this is how many decimal numbers there are. So there are a lot of decimal numbers and some of them go on forever. One over three goes on forever, but repeats itself so it couldnt of doesnt count. The square of two goes on forever and never repeats itself. Its an irrational number and its not a fraction. One over three is a fraction. So pi is another number that goes on forever and if you count all of those and you put them in with the ragsal ones, those are the real numbers and there are so many irrational numbers theres actually more of them. Another way of saying that, if you had a Hilbert Hotel where all the room numbers were decimal numbers that go on forever, think about how long it would take to ask for your key, but you wouldnt be able to evacuation that into a normal hotel if there were a fire. Theres no way of doing it. What that tells us there are more irrational numbers than rational numbers. The irrational numbers, because its a big i remember infinity, its called an uncountable infinity. Counting is 1, 2, 3, 4, 5. You can count something you can put them in the hotel. If you cant, because they dont fit, its called uncountable and by the way there are actually more irrational people than rational people. [laughter] now, this is quite a lo the to take in. Id like to have a music cal interlude now. One of my favorite songs, infinite heaven, and robert louis stevenson, id like to think he knew about infinity, uncountable angel stars, and from the whole numbers. Id let you ponder these things and infinity has inspired many people, including poets and musicians. [applause] well talk about dimensions now. Dimensions are one of my favorite things because i do high dimensional theory. And i think in dimensions all the time. Weve all been thinking about infinite dimensions without realizing it. And just like a man who had been speaking in prose all of his life without realizing. There are three dimensions in the world or are there . And does fourdimensional space exit . If we cant count time in fourdy mentio four fourdy four dimensions. If youre going to meet with somebody somewhere, you need to tell them where in three dimensional space where. And if you go at a different time you wont see them because youre in a different spot in space time. Another way to think about it, to escape from two dimensional space you need to use the third dimension. In order to climb on the stage, which is escaping from the dimensions of the floor i have to climb up the steps. If someone tried to hem me in with a fence, i could escape. If i was in a room i could escape by travelling in time. I could time travel to yesterday when i wasnt looked in a space and walk out of the space and come back to today and i would be fine. There are plenty of fictional stories and books and movies people do that and i like that as a way of thinking about the Fourth Dimension. And music, which could be considered to be a Fourth Dimension because you sort of escape from where you are when you listen to music. So, if youre stuck on a cramped train with no air, and listen to music, it takes you away and it takes you there. Its not clear if you truly escape, but what a four dimension would be. Here is a four dimensional krub. Lets think about the three dimension aal cube for a second taking a pair of two dimensional squares and joining them up. If we take a pair of three dimensional cue cubes and join them up, there are four. There are four pairs of dimensional spaces. Theres one pair and another pair of cubes inside it, here is another pair of cubes inside it does that really exist . Not clear. It does exist in our brains and it can exist in our brains even if it cant exist in the world. Thats another thing i love about math. Were only limited by what we can do in our brain. The world is much more limiting, i think. Another type of dimension, if we can about when we go to the doctor and take our vital statistics. Im all right with my Blood Pressure because my Blood Pressure and pulse are quite good. I never weigh as little as i want to, but they can take your height, weight, two numbers there. And if you put that on a graph you would need a six dimensional graph. Thats six dimensional space right there. My oh, i was about to say my and then pops up. Lets talk about criteria instead. Every time you evaluate something according to criteria, you think about dimension you want to go to. You can think about what do i want for dinner, what food they have, the service, ambiance, how much they cost and what time theyre open. If you tried to plot all of those things, youd already be thinking about that was six dimensions or something. And sometimes they merj into merge into. And so actually the criteria them selves fit on a continuum not just things that are separate so its kind of an infinite dimensional space of criteria, and thats why i think weve been thinking in infinite dimensions all of our lives. My arm only seems to move in three dimensions. But if you want to specify exactly how my harm is moving, one coordinate for the rotation of my wrist and another for how my wrist is flapping. And another how my forearm is flapping and another how the forearm is rotating and how far and back, and also how it straps. I think that might be in seven coordinates right there. And really, my arm is moving in seven dimensional space. Youll need to look at this if theres sunscreen on his book. You reach down your back here, oh, theres a mic there. And then you have to do this and move your arm here and hopefully those two points are together so you dont end up with a sunburn on your back. If the seven dimension space of my arm theyre far apart and this becomes relevant when people study robotic arms. They have a number of hinges and say youre using a robotic arm for keyhole surgery. You wont need to know whether its here in three dimensional space. But its no good if your arm had to move like this. Youll have to study the sevendy mentional state. Thinking back to the criteria, if we compare ways of going on holiday then we end up with what is a high dimensional space, thats how we could say we can compare getting to places where were going. Thats one way, and we can have different ways of getting there and we can compare the ways of getting there and maybe we could go by plane or we could go by train. We could say that train is cheaper and takes longer, plane is quicker, but more expensive. Now we compare the way of comparing. Do we care more about spending money or time. And we can xaer the wages comparing those and those, and thats kind of with dimensional space is. We could do that to compare books, as well. Sometimes people get in arguments and theyre pointless, all theyre doing is using different criteria to compare the same thing. So you might say i like this book because it has a really good plot and someone else says, yeah, this other book has better character and this person says this has a better plot. But, yeah, this has better characters. This is how arguments are on the internet and unfortunately for life as well. Its more to is it better to have a better plot or better characters. And some say i like suspense, i like these characters and you can compare the comparing. And on it goes. The moredy he thinks ms you have, the nor nuanced, if we thought more and make i could be an optimist that we could stop having shouting matches. Id like to talk about the infinitely small. And whats with a the last cookie, if i kept going forever, isnt that one over zero . That didnt make any sense. If one by infinney is zero, there are ways to define infinity one divided by zero. It doesnt make sense. This is the thing about math sometimes it is about getting the right answer, but its about exploring possible answers and what mathematicians do in research, they dream up different worlds in which different answers make sense and then they explore what else has to be true in that world and what the world looks like in order to make that answer make sense. So, in the world where one over zero is infinity, you have some ran range strange loops there, thinking of what a world would have to look like to create an answer is for me relevant to my life because he can think about what the world would have to look like in order for nobody to be afraid the mathematics. For example. Thats a dream world. I can work through it by imagining its there as well. And when you have something thats the same as itself in dinner scales, so, its like saying you could zoom in on it and it would look like itself. This is one thats famous, a triangle and in each reamending triangles, you keep doing that. Its like a loop where you keep doing the same thing and keep the triangles. And now i keep going and i couldnt go any further because my computer gave up. But you could do this with a tree. I can start with a two branches if i remember tree and make a slice like that out of each branch and now out of those and those and those and those, and i think that might be the end. Oh, no, theres one more level. You can draw your own. Im going to draw something remembering an ap lonion basket now. And although its wonky,there is is he something about drawing it and not letting the computer draw it. Draw a circle, draw five circles, teaching each other and fill in the biggest face you can find with the biggest circle to fit and keep going. You find the biggest space you can and fill it in with circles. And its quite therapeutic and meditative. The thing that looks like the biggest space, you feel like youve filled everything in and suddenly your sense of stale changes in your head and somebody looked like, oh, that was wonky. Sounded like a huge space that you fill in and you keep doing that forever and you have something that is quite although its wonky, i quite like i quite like the effect of doing it by hand. If you look, these are from the internet, there are loads and loads of beautiful xikts. Maybe ill stop now. But its addictive, now this space seems enormous. Oh, this looks enorm plus, maybe come and keep going in an empty moment. Another way is pointing two mirrors at each other, and it goes on to infinity. And the reflections and reflections and reflections get infinney small. Here is a picture of me taking a picture of me taking a picture of me, taking a picture of me forever. And the images of me get smaller inside tl. Thats why the sunday is going to do. And you might notice that this is a loop. Weve come back to loops, and theyre made by making a loop and everything something refer to itself repeatedly. One of the things i love about the mask, is the way its called an abstract. You can build math all the time. You cant build a new bird after birds, birds dont work like that. You can build a sculpture of a bird out of sculptures of birds. Because theyre ab tract enough to do that. And math an abstract enough to do that and to really close this loop. I would like to show you a poem, a basque poem translated by my friend. And demonstrates the way they make an infinite selfrestaurants, kiek off forever. And this poem refers to appointing at the fall of ikaris. And it goes to the greek myth. Ill read the home. If i had to choose a poem, id be one alden. The one which mentions the fall of ikaris, which theyre a tiny almost invisible figure falling into the sea. The old masters understood suffering well, thats wh autumn. How it takes place while someone else is eating or anticipating a window or walking dolly along. Everything urns away quite levelly and to the disaster. The pain of someones life almost never teaches someone touches one else. The english poem mentions an flemish painting. Im by the way, waiting for them to stall frlt its a good view from here, soft, without a care in the world. If he falls, his wings will hit the antenna on the roof. Theres danger in the power lines, too. If we were to join in everyones suffer, life would be impossible, but im waiting and im going to help. Ill give him a shelter when he falls like i fell. Thank you. [applaus [applause] are there any questions . If you do have questions we ask that you come up to the microphones on either side of the stage. Yeah, i was just curious when it sounds like you work with kids and teaching them interesting concepts in math and what are their reactions when you talk to them about you know, different infinities, uncountable. They scream with excitement and thats one of the reasons why i love teaching small children because theyre much less afraid of math. They havent learned to be afraid of math yet and their reactions are so pure and unfettered and some they scream with excitement and i talked to infinity with four to sixyearolds last year and came back to talk to them about Something Else and as soon as walked on the stage. Some little boy yelled infinity. Clearly remembered that from the year before. I think that childrens brains are amazing, that they can fit more things in and they havent learned to have preconceptions. Theres one boy. And i talked about infinity and a boy got upset and said there wasnt any numbers in it. And its interesting, most people are kind of put off by numbers, a lot of people who are put off by math are put off by numbers and theyre relieved when i show the things that math is that arent numbered and this little boy wanted it about numbers. Its not all about numbers. Hi. In the poem it says i felt like a chicken. I think something happened sad for the poet. And whether we should feel sad for everyone in the world. We cant because we would be overwhelmed, but we can pick one person and help them and then that will be okay. Oh. So why do you think we applaud when you finish a wonderful piece of music, but not when you do these Great Concepts and ideas . You did, i think you applauded. [applaus [applause] those numbers that go on forever without repeating, how do you have certainty that they go on forever without repeating ever . That is a great question. And its something that we do teach to, i taught it to undergraduates in first year math classes and if it cant be expressed as a fraction then it cant ever repeat itself because if it did repeat itself you could express it as a fraction. Did you follow that logic . If it were a fraction it would repeat itself and also if it repeated itself you could express it as a fraction and take the repeating part and express as a fraction. If you can prove something is not a fraction then you know it must never repeat itself. Its quite freeze i to and its easier to describe at that pii is a fraction, its fairly easy. Any more questions . If theres no more questions, we will be having a book signing in the lobby and join me for giving another round of applause for our speaker. [applause]
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