Chang. It is made possible with support from the Microsoft Corporation and with media sponsorship by kay uw. First she will speak for about 35 or 45 minutes. After that we will move into a time for your questions. If you have a question, please use the microphone on either side of the stage and make sure your questions are concise and in question form. After questions, she will sign copies of her latest book beyond infinity and expedition to the outer limits of mathematics. Books will also be available for sale at that time. Then, you should come back, april is packed. We have the biology panel on the future of health. We have paul hawken with 100 ways to reverse Global Warming and the exploration of artificial intelligence. There is so much going on and you can find all of our programming at townhall seattle. Org. Finally, i would like to thank the members in our audience tonight. Your contributions allow us not only to create dynamic programming but to make sure its accessible to everybody in our city. Members, thank you. This is your townhall and we are grateful to be a part of this with you. Now it is my pleasure to introduce to you our speaker this evening. She is a scientist and resident at the school of 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. In fact, her latest Youtube Video has been viewed over 8 million times. In 2015, she joined Steven Colbert to explore the mathematical wonders of puff pastry. Please give me a warm town help welcome. [applause] thank you very much. Thank you for that wonderful introduction. Thank you to the townhall seattle for inviting me back to talk about this book. I had a wonderful time here when i came here couple years ago. Thank you all for joining us to talk about infinity and what lies beyond infinity. I was once called in to arbitrate the dispute between a 4yearold nephew and his best friend about whether infinity was a number or not. His best friend said infinity is not a number because my daddy said so and my daddy is a scientist so he knows everything. My nephew, sensibly focused on proving that his daddy didnt know everything by rattling off batman facts that his daddy didnt know. I love the fact that infinity is something children ask about and that adults can answer, and for me that is a wonderful place to explore what goes on in the world, what goes on in our mind and to explore and nurture curiosity in children. It is amazing and wonderful to me that it took mathematicians thousands of years to pin down a way of talking about infinity that would still make sense when you scrutinize it with logic. Unfortunately, there are many arguments in the world these days that do not make sense, even when you dont scrutinize them with logic. Certainly, when you do scrutinize them with logic, and the whole thing with math is to construct arguments that hold up under logic. That is what we will look at today. There are infinitely many things i could say about infinity so id have to pick some and im going to talk about paradoxes in the infinitely big and we will touch on what infinity is. Well talk about dimensions and finish by talking about the infinitely small. One of my favorite things and they are how i think icebergs encountered infinity. As if i grew up in a house that had a chimney in the middle with a fireplace in the idea was all the other rooms went around the fireplace and were heated by the fireplace. The thing is, my sister like to chase each other around the fire place. Because it was a loop we could basically chase each other infinitely far around the house. It was like we had an infinite house. Although we could change direction and have hysterics as we crashed on the other side. This is how we get access to infinity through loops even though our world is infinite. My Favorite Computer program is one i wrote when i was five. My mother taught me to Program Computers when i was five and i was very good at it and i never got any better so i still program like a 5yearold. The thing is, i was so entranced by my Favorite Program that i never got bored of it. The thing is, im not easily bored which has its down side because i never got any better because i could always do the same program over and over. It was programmed in basic. Ten hello. Twenty go back. I would go over and over. I think this was my first appearance of infinity in a loop. What is infinity . Well my friend is a translator and she tells me in basque, infinity is the same word as 11 which kind of indicates that beyond ten it might as well be infinity. This id is borne out by my friend sally who one day was counting, she was doing home a jam and she posted this on facebook. Somehow beyond ten is lots. In my work in higher dimensional category, higher means anything higher than three so in higher dimensional category sickles one, two, three, lots. After three you might as well be doing infinity. It depends what you like about numbers in what situation you are in to determine what counts as big. Another way we can get loops is if we sting around and we probably all know this one. Id like to try to see if we can sing it. Now maybe if we start over here, everyone over here can go first and everyone over here can go second. Everyone over here can go third and everyone over here can go forth and we will see how we do. Lets start here. Row, row, row your vote row, row, row your vote row, row, row your vote merrily merrily merrily life is but a dream. Merrily merrily merrily merrily life is but a dream. Yeah, we could go on forever. There is one incident involving my nephew where my sister and i were walking him home and he was screaming and the only thing that would stop him from screaming is if we sang row, row, row your boat in around and kept going. So lets see what happened. You sang row, row, row your boat and then the next people started and the next people started and im happy you carried on it didnt just stop at the end. That means when you got to the end which is here, you started again and the other people started again in the green people started to again, and the orange people started again and we kept going. We made this loop. If you look at the four lines in each section, they are the same for, they just rotate around the four groups of people. This is the infinite loop of rope, row, row your boat. We could keep going on forever because its a loop and we can go around as many time is we like. This is another way we get to access infinity. Lets talk about paradoxes. If we think about what infinity is and were not careful about it, we run into some issues and there are things you can do when you run into problems. You can run away from it or you can investigate it. If you are a mathematician, what you tend to do is investigate why its there rather than running away from it. Here is one i really like which is called hill built hotel. You imagine you have an Infinite Hotel. Your Infinite Hotel has rooms one, two, three, four, five, six, seven, eight. He goes on forever. This is not possible in real life. The great thing is out math is that its not possible in real life. Thats what i like about it. In real life there are all sorts of restrictions like time and money and physical stuff. With math the only restriction is logic. You can have an Infinite Hotel and see what happens. You can have a full hotel and see what happens when one new guest arrives. Then you say how about i just move the person in room two into room one and i move the person in room two into room three and the person in room three in two ruroom for and so on and they never run out of rooms because it goes on forever. That means room one is empty and i can put the extra person in room one. This is something you cant do in a finite hotel. It wouldve changed the nativity theory. [laughter] these products go back more than 2000 years. There are about motion and how we understand motion. One of them said achilles can never be the horses in a race. Achilles is very fast but they said im going to challenge you to a race, i just want a head start. Not too big. The point is by the time, supposing the horses start here, then by the time achilles has arrived at this black line the tortoise has gone forward a little bit. By the time achilles has got to there, the tortoise has got some further. That means the tortoise is always going to be ahead of achilles. That doesnt seem quite right because probably the tortoise is going to be beaten by achilles. Theres a wonderful video on youtube where there is an actual race with a bunny against a tortoise. Its perfect because the bunny goes fast and the tortoise is going slow and then the bunny goes the wrong way and the tortoise actually wins. I love it. The next paradox says you cant eat all your chocolate cake because you eat half of it and then you have to eat half of whats left and then you eat half of whats left and half of whats left and so on and there would always be a bit left. There would always be half of whats left left so you would never finish your chocolate cake. And yet, i do finish my chocolate cake. The last thing is that nothing is moving because if you take a photo, if we always around and someone takes a photo, in the photo no one will be moving because it only captured one instance and in that instance no one is moving so how is it that we are moving. These are things where some form of logic has violated what we understand about the world. Some people say math is dumb. Other people say we should explore math further to see how we can resolve this using logic. I will get to some of that later. Here is the solution to all of this 2000 years later. Thats what calculus is about. You might think it is a thing invented to torture people in school. A lot of people say i was okay with math until calculus. Some people say i was okay with math until times table. I have sympathy with both of those things. Calculus is there to deal with infinity and the small things. Although you may think infinity is far away from this world, actually considering infinity and how to think about it is what led to the development of calculus which 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 way that the strings were, everything is controlled by calculus. Without that understanding, we wouldnt have any of these technological developments. That is kind of a use of infinity. Im wary of talking about how useful math is because i dont like to think its a selling point. It would be like saying i would like you to meet my friends, they are very useful. Math is useful, but often its the fact that its fascinating, not that its useful. Its great that its useful as well. Here is a picture of me eating cookies. The first cookie is one tenth of the whole dough. The next is one tenth of whats left and the next is one tenth of whats left and so on. The end one ive eaten. They are going into my mouth. So now i talk about infinitely big things a little bit. What is infinity . Is it a number . Well lets think about some big numbers. What does a million look like . In thinking about this and im thinking ive never seen a million of anything. Theres a great Childrens Book called 1 million jellybeans. No, how many jellybeans but i dont want to spoil it, but the last page is a picture of 1 million jellybeans and its really cool because you have to unfold it because its so big. Heres ten and heres a hundred, heres 10000. I couldnt make a million dots because my computer gave up. It said you dont have enough memory because i have a pathetic computer. Infinity is bigger than all numbers. Its bigger than how many people there are on earth and its more than how many germs or molecules or, we dont have an infinite number of things on earth, but this doesnt tell us what infinity is. It only tells us what infinity isnt. It would be like saying yes youre taller than everyone you know but it doesnt tell you who it is. What is infinity . We can also think about lines of symmetry on a circle. I was helping firstgraders with symmetry. We did a triangle and a hexagon in an octagon and one of them said an octagon has eight sides because art stands for octopus. I gave them a circle and wonderedrewthis line in these ld wonder these lines and one of them went, theres hundreds of them and the other one went theres millions of them and the other one went you could spend your whole life drawing them and you wouldnt finish in the other one just colored in the whole circle ive drawn them all now and i went home and i said what just happened. There are infinitely many lines of symmetry on a circle. We do have infinitely many things as long as theyre not physical things. So its also a way of winning arguments. Some of you might say im right times ten or im right times a hundred or im right times infinity, but what if somebody said im right times i infinity times two. What if Infinity Plus one equals infinity because infinity is the biggest thing. We coulds then subtract infinity from both sides of the equation and we would get one equals zero which is not so great. We could do Infinity Plus two equals infinity and we would get two equals zero and if we keep going like this we will get everything equals zero. We could even suggest that Infinity Plus infinity and we subtract infinity of both sides we get infinity equals zero. Now everything equals zero. Now we are in the zero world. Thats okay. Thats a perfectly valid mathematical world. Just not the one we live in. You cant do much and fight it. Its like the world of candy when i was little because i was allergic to food coloring so i couldnt have candy. No matter how much candy everyone gave me, i still had zero candy. A very sad world to live in. Infinity is not an ordinary number. You cant manipulate it, while you can, you can do anything in math, its just that if you call the contradiction you end up in the zero world. You can say infinity is an ordinary number but only if youre in the zero world. What is a . Its not a real number. The real numbers are the ones that fit on this line. They are the whole numbers. In between number all the real numbers but maybe theres another kind of thing it is. Mathematicians have developed this in its an ordinal or cardinal number. Its the number of things there are in something. You find some welldefined, familiar objects and you say oh like this, this is ten. This is how we tell our child what ten is. This is five. We say five and five is ten and we can do infinity like that. Infinity is how many numbers there are. Lets be careful about that. Its how many whole numbers there are. The reason is this is only the first infinity because there are more infinities beyond that because there are bigger sets than all the whole numbers. What does that possibly mean. Well, lets think about this. The first infinity is how many guests fit in the hotel. Remember, we could fit a new guest in like that but they still fit in the hotel. That seems to be telling us that Infinity Plus one is still the same number of infinity because we can still fit it in the same hotel, but it kind of depends. It depends whether you care about the hassle that you have caused many people. Imagine you check into a hotel and they said excuse me you have to move rooms. Thats a hassle. If you care about the hassle, one plus infinity is not the same as Infinity Plus one because one plus infinity means one guest arrives first and then infinity gas. In that case you put the first guest in room one and the next in room two and no one had to move. But if you put Infinity Plus one, then you put everyone in and you have to put one more person and that causes everybody to move which is a hassle. Infinity plus one is then bigger than infinity. This is what shakespeare knew because in the taming of the shrew he said i have no more to say so bid farewell beyond forever and a day. He didnt say a day and forever. Forever and a day, you have to wait for forever and then a day after that. So, there is a bigger infinity than the whole numbers. Theres a bigger infinity than the number of people in the hotel. This is how many decibel numbers there are. There are a lot of decibel numbers and some of them go on forever. One of the three goes on forever but it repeats itself so some doesnt help. Its called an irrational number which means its not a fraction. One over three is a fraction. Pie is another decimal number thats very famous but goes on forever and never was pizza itself. Its irrational. If you count all of those ones and you put them in with the rational ones, those are all the real numbers and there are so many irrational numbers that theres actually more of them. Another way of saying that is if you have the 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, you wouldnt be able to evacuate that into a normal hotel. It wouldnt fit. Theres no way of doing it. What thats telling us is there are more irrational numbers than rational numbers. The irrational numbers, because its a bigger infinity, its called an uncomfortable infinity because counting is going 12345, if you can count something then you can put them in the hotel. If you cant because they dont fit its called uncomfortable. By the way, there are also more irrational people than rational. Now thats quite a lot to take in so id like to have a musical interlude now. One of my favorite songs, the infinite shining heavens by von williams. I like to think that he knew about this as well because he says uncountable angels so he clearly thanks the next order of infinity up from the whole numbers so i will now let you ponder these things a little bit and we will think about the fact that infinity i has inspired may people including poets and musicians. [applause] we are going to talk about dimensions, dimensions are one of my favorite things. I think in high dimensions all the time and i think actually weve all been thinking in infinite dimensions all our lives without realizing it. Lets think about how many dimensions there are in the world. There are really only three dimensions in the world. Does four dimensional space exist . I can. To time because one of his. [inaudible] if we count time then we get to understand how gravity works. Spacetime is a very famous four dimensional space and you can think of it as this. If you are going to meet up with someone you need to tell them where in threedimensional space but you also need to tell them when because if you dont you wont see them because youre in a different spot in spacet