Transcripts For CSPAN2 Eugenia Cheng Discusses Beyond Infini

Possible with support for the Microsoft Corporation and with media sponsorship. First this evening, the other will speak for 35 or 45 minutes and then will move into questions. If you have a question please use the microphone on either side of the stage and make sure your question is concise and in question form. After questions virginia will sign copies of her latest book, beyond infinity and addition of the outer limits of mathematics. Books will be available for sale. Then you should come back and aprils packs, we have the Seattle Institute of biologys panel on the teacher of health. We have paul hawken with 100 ways to reduce mobile warming and artificial and emotional intelligence. You can find all of our programming at townhall seattle. Org. Finally, i would like to thank the members and their audience, your contributions allow us to create dynamic programming and make sure it is accessible to everybody in our city. So thank you. This is your town hall, we are grateful to be a part of this with you. It is my pleasure to introduce to you our speaker this evening. Virginia is a scientist in residence at the school of Art Institute in chicago, an honorary fellow of the university of sheffield. Her popular youtube videos use math to explain everything from the perfect class of want to the best way to share cake. It has been viewed over 1 million times. Her latest video has been viewed over 8 million times. In 2015 she joined Stephen Colbert on the late show to explore the mathematical wonders of possible pastry. Join me and giving me a warm welcome to virginia chang. [applause] thank you for that wonderful introduction. Thank you to townhall seattle for inviting me back to talk about the book. Had a wonderful time when i talk about my previous book a few years ago. Thank you for joining us to talk about infinity and what lies beyond infinity. I was once called into arbitrated dispute between my 4yearold nephew and his best friend about whether infinity was a number not. His best friend said infinity was not a number because my daddy said so my dad is a scientist that he knows everything. My nephew sensibly focused on that the his daddy didnt know everything by reeling off batman facts. I love the thing that infinity is something children ask about an adults can answer. Thats a wonderful place to explore what goes on in the world and in our minds. And to explore and nurture curiosity in children who took mathematicians thousands of years to talk about infinity when he scrutinize it with logic. There are arguments in the world that didnt make sense. Even when you dont scrutinize them with logic. But certainly when you do scrutinize them logic. The whole point of math is to construct arguments that hold up under logic. Thus what will look at today. There are infinitely many things i could say but ive had to pick some. Im going to talk about paradoxes, the infinitely big, i like to touch on what infinity is and talk about dimension and finish by talking about the infinitely small. One of my favorite things, how i think i first encountered infinity. I grew up in a house that had a chimney in the middle with a fireplace. The idea was the other rooms went around the fireplace overheated by the fireplace. My sister and i like to chase each other and because the rooms are in the loop we could chase each other infinitely far around the house. As if we had an infinite house because we could change it direction. But is there way we get access to infinity even though our world is finite . One of my Favorite Computer programs is one i wrote when i was five. My mother taught me to compute programs when i was at five and i was good at it and i never got better. The thing is i was so struck by my Favorite Program that i never got bored of it. Im not easily bored which has its downside because i never got better at programming because i could always do the same program over and over again it was programmed in basic which dates me. It goes ten hello, 20 go to ten, and then you hit run and you get this. And i could do this over and over again. And now we have an infinite loop. This is probably along with chasing my sister around the house was my first appearance of infinity in the loop. What is infinity . Well, my friends is a translator and she tells me and bask infinity is the same word as a 11. And that indicates that beyond tend to medicine be infinity. This ideas borne out by my friend sally who is counting and making homemade jam and she had three dogs in 2013, and a lot from 2015 so somehow beyond ten is loss. In my work and higher dimensional category theory higher means anything higher than three. So goes one, two, three lost and after three minus will be doing infinity. It depends what you like about numbers. Another way is if we sing around and we probably all know this one. So i like to try and see if we can sing it altogether. Now maybe if we start over here everyone over here can go first and everyone over here can go second, third, and fourth. Lets start here. Row, row, row your boat, row, row, row your boat, row, row, row your boat. Row, row, row your boat merrily, merrily, merrily, life is but a dream there is one incident involving my nephew when we were walking my nephew home and it was raining and he was screaming. The only thing stopped him from screaming is if we saying row your boat and around and didnt stop. Heres my depiction of what we just did. So the next people start here in the next people start here, but im happy that you spontaneously carried on and did not stop at the end, which means when you got to the end, which is here you started again and the other people started again when the orange people also started again and then we kept going. We made a loop. If you look at the four lines in each section they just rotate around the small groups of people. So this loop is the infinite loop of row, row, row your boat and we could keep going forever because its a loop and we can go around it. So this is another way we can access infinity. Lets talk about paradoxes. If we think about what infinity is in a careful, we run into issues. Theres a few things you can do, you can right away fo from its r say, this looks interesting, i think i will investigate why this paradox happens. If your mathematician you investigate why the paradox is there. So, heres what i like. Imagining your head that you have an Infinite Hotel. So your Infinite Hotel has a rooms one, two, three, four, all the way on forever. This is not possible in real life but the great thing about math is that it is not real life. Some people think its terrible but i like it because it means you can do things that are not possible in real life. There are all sorts of restrictions like time and mon money. And that the only restriction is logic. See can have an Infinite Hotel and see what happens. You can have a full hotel and see what happens when one new guests arrive on the need to go, how about i move the person in room two into the room wanted to rent to and the person in room three to room four and so on. And they never run out of rooms to move into because it goes on forever. That means room one is nt and i can put the extra person in room one. This is something you cannot do in a finite hotel. But it would change the nativity story. But these go back more than 2000 years. They are about motion and how we understand motion one of them said that achilles can never beat the horses in the race but they said im going to challenge you to a race i just want to head start. Not too big. And so by the time supported they started this line, by the time achilles has arrived at this blackline the tortoise has gone forward a bit and maybe by the time achilles has got to their the tortoises got further. So that means the tortoise is always going to be ahead, but that doesnt seem quite right because probably the tortoises going to be beaten by kellys, theres wonderful video on youtube where theres an actual race with a bunny against the tortoise. The bunny and and then the bunny goes, oh, oh. And eventually everyone screaming at the bunny to go that way and he never does. Its brilliant, i love it. Next paragraph says that you cannot eat all your chocolate cake because if you eat half of it i knew it half of whats left and then you eat have of whats left and so on. So youll never finish your chocolate cake. And yet, i do finish my chocolate cake. On the last one is that nothing is moving because if you take a photo, if we all waive around, in the photo nobody will be moving because it only captures one instant. In that instant no one is moving so how is that we are moving . So some form of logic has violated what we understand about the world. Some people say that math is done, other people say we should explore math further to resolve this using magi logic. The solution to all of this came 2000 years later and thats what calculus is about. You might think it was invented to torture people in school. People say i was okay with math until calculus. For some people say that until times table. I have sympathy with both. Calculus is there to do with infinitys although you may think that 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 electricity, everything that has a curve in it. The curve of the piano, the way the string warts, everything is controlled by calculus. So without the understanding of infinity we would not understand this. My bit wary of talking about how useful methods, because i dont like to think that its a big selling point, it would be like saying, i would like you to meet my friend, theyre very useful. So map is useful, but often is the fact that its fascinating out that its useful, its great that its useful as well. Heres a picture me eating the cookies. I made Chocolate Chip Cookie dough on the first one is one tenth of the doubt. In the next was one tenth of whats left and then one tenth of whats left and so on. The end ones ive be eaten into my mouth. So now lets talk about infinitely big things a little bit, what is infinity, is that a number . Lets think about big numbers, which is 1 million look like . And i think weve never seen a million of anything but theres a great Childrens Book called how many jellybeans. I dont want click bait either the last pages what the pitcher of 1 million jellybeans and its really cool because you have to unfold the because its so big. Here stan, heres 100, heres 10000, cannot make a million dots because my computer gave up. Because i had a pathetic computer. But infinity is bigger than all numbers. Its bigger than how many people honor and its more than how many ants, germs, cells or molecules. We dont have an infinite number of things on earth. This does not tell us what infinity is, it only tells us what infinity isnt. Which would be like say youre taller than anyone you know but it doesnt tell you who he is. What is infinity . We can also think about there are infinitely many things on earth, but how about lines of symmetry in a circle. I was once helping some firstgraders with symmetry. We did a triangle, square and so on. One of them said, an accident has eight sized kazakh stands for octopus. So i eventually gave them a circle and one of them drew a line and then one drew these one of them said, theres hundreds of them. And one said theres millions of. And the other one said you can spend your whole life drawing them and you would not finish. On the other one just colored in the circle and said draw them all now. And i went home going, what just happened . But there are infinitely many lines of symmetry on the circle. We had infinitely many things and not all physical things. Infinity is way of winning arguments because im right times ten, time so hard im right times infinity. But then what if someone says im right times to infinity. Is it bigger . What happens if you think that Infinity Plus one equals infinity because infinity is the biggest thing . We could subtract infinity from both sides of the equation and we would get one equals zero which is not so great. We could then do Infinity Plus two equals infinity and then well get to equal zero. If we keep going will get everything equal zero. We could even suggest that to infinity is Infinity Plus infinity and we subtract it from both sides that we get infinity equal zero. Now everything equal zero. Now are in the zero world. Thats a perfectly valid mathematical world just not one we live in. Its like the world of candy when i was little. I was allergic to food coloring so i can have candy. No matter how much candy anyone gave me a still had zero candy. Very sad world to live in. Infinity is not an ordinary number. You can if you cause a contradiction you end up in the zero world. You can say infinity is an ordinary number but only fear the zero world. So what is it . Its not real number. The real numbers on that fit on this line. The whole numbers in between them are the real numbers. Maybe theres another kind and mathematicians have developed it and its called a cardinal number and its just the number of things there are in something, say find some very welldefined familiar set of objects and you go like this, this is ten this is how we tell it child what tenants. This is five and then we say five out of five is five, 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 on earth does that mean . Well, lets think about this. The first infinity is how many guests fit in the hotel. So remember we could fit a new guest in but they still fit in the hotel so that seems to be telling us that Infinity Plus one is still the same number because we can still fit it in the same hotel. But it depends on whether you care about the hassle you have cost infinitely number people. Imagine you have checked into a hotel and they say you need to move rooms. Thats a hassle. So if you care about the hassle one plus infinity is not the same as Infinity Plus one. One plus affinity means one guest rides first. In that case you just put the first guest in room one and the infinity guest at room to a nobody has to move. Infinity plus one means you put the infinity guest in first in the new guest arrives and then you have to move everyone up a room so you cause everyone hassle. That is different. If youre doing that kind of thing that Infinity Plus one is bigger than infinity. This is what i think shakespeare means. In the taming of the shirt he said i have no more to say but farewell forever and a day. He evidently thought it was longer than forever. Because you have to wait for the whole of forever and then another day after that. There is a bigger infinity than the whole number. Bigger infinity than the number of people in the hotel. This is how many decibel numbers there are. There are a lot of decimal numbers. Some go on forever. One over three goes on forever but repeats itself. The square root of two goes on forever and never repeats itself which is because its called an irrational number which means its not a fraction. So, pie is another decimal number thats very famous that goes on forever. And never repeats itself. Its irrational. If you count all of those and you put them in with the rational those are all the real numbers. There are so many irrational numbers there are more of them. So if you had a hotel where all of the room numbers were decibel numbers that went on forever, think about how long it would take but you wouldnt be able to evacuate into a normal hotel if there is a fire. Was that telling us is there are more irrational numbers than rational numbers. Irrational numbers because its a bigger infinity is called uncountable infinity. Counting is going one, two, three, four, five. If you can count something you can put it in the hotel, if you cant its called uncountable. By the way, there are more irrational people than rational people. Thats a lot to take inside like to have a musical interlude now. One of my favorite songs. I would like to think that he knew about different infinity because he said, uncountable angel stars. He thinks that is the next order of infinity up from the whole number. So i will now let you ponder these things a little bit will think about the fact that infinity has inspired many people including poets and musicians. Mocha ma[applause] were going to talk about dimensions now. When my favorite things because i think in high dimensions and i think actually we have been thinking and dimensions all her life. So lets think about how many dimensions there are there are really only three in the world. Einstein made four dimensionals because one of his things was if we count time is a Fourth Dimension the kids understand how gravity works in a better way. Spacetime is a famous four dimensional space. If youre going to meet up with somebody somewhere, you need to tell them where in threedimensional space to me about. So to this day from twodimensional space so to climb onto the stage then i have to climb up some steps. If i can fly into the third dimension i could escape. Someone blocks me in i would be closed in three dimensions but i could escape by traveling in time. I could time travel to yesterday when i wasnt locked in space. Then i could walk out and come back to today. There are plenty of fictional stories and books and movies where people do that. I like that as a way of thinking about the Fourth Dimension. Theres also music which could be considered a Fourth Dimension because you escape from where you are when you listen to music so if youre stuck, try another no air but its a thought then here is a four dimensional cube. So, how is this a four dimensional queue. Lets think about the threedimensional cubit. A threedimensional cube is made by taking a pair of two dimensional squares and joining them. If we take a pair of threedimensional cubes and joined them up, that is a four dimensional queue. There are four pairs of parallel threedimensional faces. Heres another pair inside it and here is another pair of cubes. So does that really exist . Its not clear it can exist in our brains even if it cant exist in the world and thats another thing i love about math because were only limited by what we can do in our brain the world is much more limiting than that. So another price of dimension is if we just think about what would go to the dr. And they take our Microsoft Corporation<\/a> and with media sponsorship. First this evening, the other will speak for 35 or 45 minutes and then will move into questions. If you have a question please use the microphone on either side of the stage and make sure your question is concise and in question form. After questions virginia will sign copies of her latest book, beyond infinity and addition of the outer limits of mathematics. Books will be available for sale. Then you should come back and aprils packs, we have the Seattle Institute<\/a> of biologys panel on the teacher of health. We have paul hawken with 100 ways to reduce mobile warming and artificial and emotional intelligence. You can find all of our programming at townhall seattle. Org. Finally, i would like to thank the members and their audience, your contributions allow us to create dynamic programming and make sure it is accessible to everybody in our city. So thank you. This is your town hall, we are grateful to be a part of this with you. It is my pleasure to introduce to you our speaker this evening. Virginia is a scientist in residence at the school of Art Institute<\/a> in chicago, an honorary fellow of the university of sheffield. Her popular youtube videos use math to explain everything from the perfect class of want to the best way to share cake. It has been viewed over 1 million times. Her latest video has been viewed over 8 million times. In 2015 she joined Stephen Colbert<\/a> on the late show to explore the mathematical wonders of possible pastry. Join me and giving me a warm welcome to virginia chang. [applause] thank you for that wonderful introduction. Thank you to townhall seattle for inviting me back to talk about the book. Had a wonderful time when i talk about my previous book a few years ago. Thank you for joining us to talk about infinity and what lies beyond infinity. I was once called into arbitrated dispute between my 4yearold nephew and his best friend about whether infinity was a number not. His best friend said infinity was not a number because my daddy said so my dad is a scientist that he knows everything. My nephew sensibly focused on that the his daddy didnt know everything by reeling off batman facts. I love the thing that infinity is something children ask about an adults can answer. Thats a wonderful place to explore what goes on in the world and in our minds. And to explore and nurture curiosity in children who took mathematicians thousands of years to talk about infinity when he scrutinize it with logic. There are arguments in the world that didnt make sense. Even when you dont scrutinize them with logic. But certainly when you do scrutinize them logic. The whole point of math is to construct arguments that hold up under logic. Thus what will look at today. There are infinitely many things i could say but ive had to pick some. Im going to talk about paradoxes, the infinitely big, i like to touch on what infinity is and talk about dimension and finish by talking about the infinitely small. One of my favorite things, how i think i first encountered infinity. I grew up in a house that had a chimney in the middle with a fireplace. The idea was the other rooms went around the fireplace overheated by the fireplace. My sister and i like to chase each other and because the rooms are in the loop we could chase each other infinitely far around the house. As if we had an infinite house because we could change it direction. But is there way we get access to infinity even though our world is finite . One of my Favorite Computer<\/a> programs is one i wrote when i was five. My mother taught me to compute programs when i was at five and i was good at it and i never got better. The thing is i was so struck by my Favorite Program<\/a> that i never got bored of it. Im not easily bored which has its downside because i never got better at programming because i could always do the same program over and over again it was programmed in basic which dates me. It goes ten hello, 20 go to ten, and then you hit run and you get this. And i could do this over and over again. And now we have an infinite loop. This is probably along with chasing my sister around the house was my first appearance of infinity in the loop. What is infinity . Well, my friends is a translator and she tells me and bask infinity is the same word as a 11. And that indicates that beyond tend to medicine be infinity. This ideas borne out by my friend sally who is counting and making homemade jam and she had three dogs in 2013, and a lot from 2015 so somehow beyond ten is loss. In my work and higher dimensional category theory higher means anything higher than three. So goes one, two, three lost and after three minus will be doing infinity. It depends what you like about numbers. Another way is if we sing around and we probably all know this one. So i like to try and see if we can sing it altogether. Now maybe if we start over here everyone over here can go first and everyone over here can go second, third, and fourth. Lets start here. Row, row, row your boat, row, row, row your boat, row, row, row your boat. Row, row, row your boat merrily, merrily, merrily, life is but a dream there is one incident involving my nephew when we were walking my nephew home and it was raining and he was screaming. The only thing stopped him from screaming is if we saying row your boat and around and didnt stop. Heres my depiction of what we just did. So the next people start here in the next people start here, but im happy that you spontaneously carried on and did not stop at the end, which means when you got to the end, which is here you started again and the other people started again when the orange people also started again and then we kept going. We made a loop. If you look at the four lines in each section they just rotate around the small groups of people. So this loop is the infinite loop of row, row, row your boat and we could keep going forever because its a loop and we can go around it. So this is another way we can access infinity. Lets talk about paradoxes. If we think about what infinity is in a careful, we run into issues. Theres a few things you can do, you can right away fo from its r say, this looks interesting, i think i will investigate why this paradox happens. If your mathematician you investigate why the paradox is there. So, heres what i like. Imagining your head that you have an Infinite Hotel<\/a>. So your Infinite Hotel<\/a> has a rooms one, two, three, four, all the way on forever. This is not possible in real life but the great thing about math is that it is not real life. Some people think its terrible but i like it because it means you can do things that are not possible in real life. There are all sorts of restrictions like time and mon money. And that the only restriction is logic. See can have an Infinite Hotel<\/a> and see what happens. You can have a full hotel and see what happens when one new guests arrive on the need to go, how about i move the person in room two into the room wanted to rent to and the person in room three to room four and so on. And they never run out of rooms to move into because it goes on forever. That means room one is nt and i can put the extra person in room one. This is something you cannot do in a finite hotel. But it would change the nativity story. But these go back more than 2000 years. They are about motion and how we understand motion one of them said that achilles can never beat the horses in the race but they said im going to challenge you to a race i just want to head start. Not too big. And so by the time supported they started this line, by the time achilles has arrived at this blackline the tortoise has gone forward a bit and maybe by the time achilles has got to their the tortoises got further. So that means the tortoise is always going to be ahead, but that doesnt seem quite right because probably the tortoises going to be beaten by kellys, theres wonderful video on youtube where theres an actual race with a bunny against the tortoise. The bunny and and then the bunny goes, oh, oh. And eventually everyone screaming at the bunny to go that way and he never does. Its brilliant, i love it. Next paragraph says that you cannot eat all your chocolate cake because if you eat half of it i knew it half of whats left and then you eat have of whats left and so on. So youll never finish your chocolate cake. And yet, i do finish my chocolate cake. On the last one is that nothing is moving because if you take a photo, if we all waive around, in the photo nobody will be moving because it only captures one instant. In that instant no one is moving so how is that we are moving . So some form of logic has violated what we understand about the world. Some people say that math is done, other people say we should explore math further to resolve this using magi logic. The solution to all of this came 2000 years later and thats what calculus is about. You might think it was invented to torture people in school. People say i was okay with math until calculus. For some people say that until times table. I have sympathy with both. Calculus is there to do with infinitys although you may think that 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 electricity, everything that has a curve in it. The curve of the piano, the way the string warts, everything is controlled by calculus. So without the understanding of infinity we would not understand this. My bit wary of talking about how useful methods, because i dont like to think that its a big selling point, it would be like saying, i would like you to meet my friend, theyre very useful. So map is useful, but often is the fact that its fascinating out that its useful, its great that its useful as well. Heres a picture me eating the cookies. I made Chocolate Chip Cookie<\/a> dough on the first one is one tenth of the doubt. In the next was one tenth of whats left and then one tenth of whats left and so on. The end ones ive be eaten into my mouth. So now lets talk about infinitely big things a little bit, what is infinity, is that a number . Lets think about big numbers, which is 1 million look like . And i think weve never seen a million of anything but theres a great Childrens Book<\/a> called how many jellybeans. I dont want click bait either the last pages what the pitcher of 1 million jellybeans and its really cool because you have to unfold the because its so big. Here stan, heres 100, heres 10000, cannot make a million dots because my computer gave up. Because i had a pathetic computer. But infinity is bigger than all numbers. Its bigger than how many people honor and its more than how many ants, germs, cells or molecules. We dont have an infinite number of things on earth. This does not tell us what infinity is, it only tells us what infinity isnt. Which would be like say youre taller than anyone you know but it doesnt tell you who he is. What is infinity . We can also think about there are infinitely many things on earth, but how about lines of symmetry in a circle. I was once helping some firstgraders with symmetry. We did a triangle, square and so on. One of them said, an accident has eight sized kazakh stands for octopus. So i eventually gave them a circle and one of them drew a line and then one drew these one of them said, theres hundreds of them. And one said theres millions of. And the other one said you can spend your whole life drawing them and you would not finish. On the other one just colored in the circle and said draw them all now. And i went home going, what just happened . But there are infinitely many lines of symmetry on the circle. We had infinitely many things and not all physical things. Infinity is way of winning arguments because im right times ten, time so hard im right times infinity. But then what if someone says im right times to infinity. Is it bigger . What happens if you think that Infinity Plus<\/a> one equals infinity because infinity is the biggest thing . We could subtract infinity from both sides of the equation and we would get one equals zero which is not so great. We could then do Infinity Plus<\/a> two equals infinity and then well get to equal zero. If we keep going will get everything equal zero. We could even suggest that to infinity is Infinity Plus<\/a> infinity and we subtract it from both sides that we get infinity equal zero. Now everything equal zero. Now are in the zero world. Thats a perfectly valid mathematical world just not one we live in. Its like the world of candy when i was little. I was allergic to food coloring so i can have candy. No matter how much candy anyone gave me a still had zero candy. Very sad world to live in. Infinity is not an ordinary number. You can if you cause a contradiction you end up in the zero world. You can say infinity is an ordinary number but only fear the zero world. So what is it . Its not real number. The real numbers on that fit on this line. The whole numbers in between them are the real numbers. Maybe theres another kind and mathematicians have developed it and its called a cardinal number and its just the number of things there are in something, say find some very welldefined familiar set of objects and you go like this, this is ten this is how we tell it child what tenants. This is five and then we say five out of five is five, 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 on earth does that mean . Well, lets think about this. The first infinity is how many guests fit in the hotel. So remember we could fit a new guest in but they still fit in the hotel so that seems to be telling us that Infinity Plus<\/a> one is still the same number because we can still fit it in the same hotel. But it depends on whether you care about the hassle you have cost infinitely number people. Imagine you have checked into a hotel and they say you need to move rooms. Thats a hassle. So if you care about the hassle one plus infinity is not the same as Infinity Plus<\/a> one. One plus affinity means one guest rides first. In that case you just put the first guest in room one and the infinity guest at room to a nobody has to move. Infinity plus one means you put the infinity guest in first in the new guest arrives and then you have to move everyone up a room so you cause everyone hassle. That is different. If youre doing that kind of thing that Infinity Plus<\/a> one is bigger than infinity. This is what i think shakespeare means. In the taming of the shirt he said i have no more to say but farewell forever and a day. He evidently thought it was longer than forever. Because you have to wait for the whole of forever and then another day after that. There is a bigger infinity than the whole number. Bigger infinity than the number of people in the hotel. This is how many decibel numbers there are. There are a lot of decimal numbers. Some go on forever. One over three goes on forever but repeats itself. The square root of two goes on forever and never repeats itself which is because its called an irrational number which means its not a fraction. So, pie is another decimal number thats very famous that goes on forever. And never repeats itself. Its irrational. If you count all of those and you put them in with the rational those are all the real numbers. There are so many irrational numbers there are more of them. So if you had a hotel where all of the room numbers were decibel numbers that went on forever, think about how long it would take but you wouldnt be able to evacuate into a normal hotel if there is a fire. Was that telling us is there are more irrational numbers than rational numbers. Irrational numbers because its a bigger infinity is called uncountable infinity. Counting is going one, two, three, four, five. If you can count something you can put it in the hotel, if you cant its called uncountable. By the way, there are more irrational people than rational people. Thats a lot to take inside like to have a musical interlude now. One of my favorite songs. I would like to think that he knew about different infinity because he said, uncountable angel stars. He thinks that is the next order of infinity up from the whole number. So i will now let you ponder these things a little bit will think about the fact that infinity has inspired many people including poets and musicians. Mocha ma[applause] were going to talk about dimensions now. When my favorite things because i think in high dimensions and i think actually we have been thinking and dimensions all her life. So lets think about how many dimensions there are there are really only three in the world. Einstein made four dimensionals because one of his things was if we count time is a Fourth Dimension<\/a> the kids understand how gravity works in a better way. Spacetime is a famous four dimensional space. If youre going to meet up with somebody somewhere, you need to tell them where in threedimensional space to me about. So to this day from twodimensional space so to climb onto the stage then i have to climb up some steps. If i can fly into the third dimension i could escape. Someone blocks me in i would be closed in three dimensions but i could escape by traveling in time. I could time travel to yesterday when i wasnt locked in space. Then i could walk out and come back to today. There are plenty of fictional stories and books and movies where people do that. I like that as a way of thinking about the Fourth Dimension<\/a>. Theres also music which could be considered a Fourth Dimension<\/a> because you escape from where you are when you listen to music so if youre stuck, try another no air but its a thought then here is a four dimensional cube. So, how is this a four dimensional queue. Lets think about the threedimensional cubit. A threedimensional cube is made by taking a pair of two dimensional squares and joining them. If we take a pair of threedimensional cubes and joined them up, that is a four dimensional queue. There are four pairs of parallel threedimensional faces. Heres another pair inside it and here is another pair of cubes. So does that really exist . Its not clear it can exist in our brains even if it cant exist in the world and thats another thing i love about math because were only limited by what we can do in our brain the world is much more limiting than that. So another price of dimension is if we just think about what would go to the dr. And they take our Vital Statistics<\/a> which i dont like thinking about. My Blood Pressure<\/a> and pulse are quite good. I never wake quite as little as i want to. They can take your weight, height and Blood Pressure<\/a>. Your pulse, age, if youre going to chart that on a graph you would need a six dimensional graph. Thats a six dimensional space right there. What about criteria instead. Every time you evaluate something according to criteria youre thinking about different dimensions. So you can think about what restaurant you want to go to for dinner. So you can think about and evaluate restaurants according to where they are, what kind of food they have, with the services like, how much they cost, what time theyre open. If you tried to plot all of those things youd be thinking that was six dimensions of something. Then sometimes they merge into each other because maybe its related to how expensive the restaurant is. So actually the criteria themselves fit on a continuum, not just things that are separate. Its an infinite dimensional space of criteria. Thats i think we been thinking of infinite directions all my life. My arm seems to move only in three dimensions, but if you think about it like this if you want to specify how my arm was moving there is one coordinate for the rotation of my wrist and another for how its flapping. Another for how my forearm is flapping and another for how its rotating. And also the rotation of my upper arm. I think that mightve been seven coordinates right there. So my arm is moving in seven dimensional space. So you need the sometimes of your putting sunscreen on your back. He reached down your back to here and then you have to do this and move your arm to this and hopefully those two points are close enough together that you dont up with some burn across your back. In threedimensional space this point in this pointer next to each other. But in the seven dimensional space my arm therefore part because i have to go around here like this. This becomes relevant when people study robotic arms because they have hinges. Say youre doing a robotic arm to do surgery then you need to know whether this thing is here in threedimensional space but its no good of the arm had to do that to move one piece inside. You need to study the seven dimensional space not the threedimensional space. Thinking back to the criteria, if we compare ways of going on holiday we end up with whats a higher dimensional space in my work in theory. We could compare the ways of getting to the place were going that is one dimension because were traveling somewhere. We could compare the different ways of getting there. Maybe we could go by train or go by plane. Then we could compare the ways of comparing. We could say train is cheaper and playing this quicker but more expensive. So now we compare the ways of comparing. Do you care more about spending money or spending time . Then we compare ways of comparing and that is kind of what infinite to dimensional space in theory is. We could do this like comparing books. Sometimes people get into arguments and often the arguments are pointless. All they are doing is using different criteria to compare the same thing. You might say i like this book because it has a good plot. And someone says this book has better characters on the senses this has a better plot this is how arguments go on the internet and in my for the moment as well. Would be more sensible to compare the criterion say is it better to have a better posture better character and then some people say i like it better plot because i like suspense or i like thinking about Human Behavior<\/a> then he can compare the ways of comparing. Then you never have to stop. The more dimensions you use the more nuance you have the way think about the world. If we all thought more infinitely dimensional than the arguments would become more nuance than just shouting matches. Maybe im being in optimist. Finally i want to talk about the infinitely small. When i am a zeno cookies, maybe the last bullet was the last wasnt the last 11 over infinity . Is at zero . If one divided by infinity zero then you could multiply everything by infinity and then what would happen . There are ways of defining infinity to be one divided by zero play doesnt make sense. This is the thing about math when its not about getting the right answer but really its about exploring possible answers. What mathematicians doing research his dream up different worlds in which different answers make sense. Then they explore what has to be true in that world in order to have that answer make sense. In a world where one over zero is infinity you have strange things in your space because its not like normal space. Thats a beautiful thing. Thinking about what a world would have to look like to create an answer is relevant to my life. I can then think about what the world would have to look like in order for nobody to be afraid of mathematics. Thats currently a dreamworld but i can work towards it by thinking of it as an imaginary in my world. Factors are where you can make things small you have things the same of itself but on a different scale. You can zoom in but it was still look like itself. Here is one thats famous. You take a at the triangle from each of the remaining you take out the triangle and then you just keep doing that. Its like a loop where you keep doing the same thing. And i keep going and i cannot get further because my computer gave up. But you could do this with a tree. So i can start with the branch for a tree and ill make up branch come out and now ill make another one, and those i think that might be the end. Theres one more level. You can try euro. Im going to trust something resembling a apollonian gasket. Theres something i like about drive it myself, heres what you do, you draw circle on the three circles inside the circle touching each other. Then you fill in the biggest space you can find with the biggest circle you can fit. And then you keep going. So that instruction repeats itself. Safe in the biggest space you can and you fill it in with the circle. Its quite therapeutic and meditative and the interesting thing is the thing that looks like the biggest space you feel like you filled everything in and then suddenly this sense of scale changes in something that previously looked like a small space starts looking like a huge space that you can fill in any keep going like that forever and then you have something that is quite fractal. Although its very wonky, i quite like the effect of doing it by hand. If you look these up by the internet there are loads and loads of beautiful pictures maybe ill stop now. So now this space likes enormous. Maybe you can all common keep going in a moment. So another way you can get fractal light thing is by pinpointing to for each other. The reflections get infinitely small. Here is a picture of me taken a picture of me take a picture of me, taking a picture of me forever. Is the drop effect. The images of me get infinitely small. You might have noticed this whole talk has gone in the loop. We have come back to loops which is what we talked to in the beginning. One of the things ive learned is how it relates to excel. You can always build more math out of math. You cant build a new bird out of birds. Birds dont work like that. You can build a sculpture of a bird sculptures of birds. Because sculptures are abstract. I think ours is abstract enough to do that. Art referred to itself i can also refer to other art. So to close this loop like to show you a poem that has been translated by my friends which demonstrate the way these references make an infinite self reference said this will actually refer to another poem that refers to the form of the fall of icarus. Im going to redo this poem. If i had to choose a poem, by wh orden. Where icarus is about a tiny, almost invisible figure falling into the sea. The old masters understood suffering well. How it takes place were someone elses eating or opening a window, or walking along. Everything turns away, walks away quite literally. But oblivious to the disaster the pain of someones life almost never touches someone else. Almost no one cares for the wounds of others. The english poem, references a painting. The painting, greek myth, who knows what. I am by the window waiting for icarus to fall. I get to good view from here is the cloud floats northward south. Without a care in the world. If he falls his wings will hit the antenna on the roof. The danger in the power lines. If we were to join in everyone suffering, life would be impossible. But i am waiting for icarus and i will help. I will collect the pieces of broken wing and give him shelter when he falls the way i fell like a chicken. Thank you. [applause] other any questions . If you have questions please come to the microphone on either side of the stage. I was curious it sounds like you work with kids. And interesting concepts in math. What other reactions when they talk about different infinities . They scream with excitement. Thats one of the reasons i love teaching small children. Theyre less afraid of math. The reactions are so pure. With all the children you get looking nervous to see if theyre okay. With children they just jump up and down physically. They scream with excitement. I came back to talk about Something Else<\/a> and some little boyd yelled until he remembered that about. I think childrens planes brains are amazing and they dont want to have preconception so much about what math is. I did a talk about infinity and then this year i talked about symmetry. This boy got upset and said there was not any numbers in it. Most people are put off by numbers and then there are relieved when i show them all the things that math is that are not numbers in this little boy wanted it to be about numbers. Its not all about numbers. In the poem and said i felt like its chicken, not sure what it means. I think it means something sadness happened about the poet and there wondering if we should feel sad for everybody if they had something sad the world and the same we cant because we be overwhelmed. But we can take one person and help them and then that will be okay. Why do you think we applied when you finish a great piece of music but not when you did these Great Concepts<\/a> . You did though, didnt you . The numbers are gone forever without repeating, how do you have certainty that they go on forever without repeating, ever . That is great question. It is something we do teach to undergraduates and first year honors math. If it cant be expressed as a fraction that i can ever repeat itself because if it did repeat itself you could ask you could make it into a fraction. If it were fraction it would repeat itself. If it repeated itself you could express it as a fraction. You can take the repeating part and use that to express the fraction. So if you can prove that something is not the fraction then you know it must never repeat itself. Its easier to prove. Its harder to prove that pie incentive fraction. The square of two is fairly easy. If there are no more questions will be having a book signing. Please join me in giving another round of applause. [applause] [inaudible] saturdays earth day, and washington, d. C. Protesters are gathering on the National Mall<\/a> for the march for science rally with speeches from scientists and activists along with musical performances. Watch live coverage at 10 00 a. M. Eastern on cspan and cspan. Org. Listen live with the cspan radio app. This week it on cspan three, saturday at 7 00 p. M. Eastern georgia tech history professor gregory noble talks about the influence of 19th century ethologists, john strings audubon and how he helped pioneer citizen science. The artistic work but also his fieldwork. He was very good at what he did and he did it with no binoculars, no field guides, no iphone apps, and the proof is in the painting. At 8 00 p. M. , lectures in history alan kelso on Abraham Lincoln<\/a> and the dred scott u. S. Supreme court decision. What is he saying . There is now no restraint, non even the restraint of popular sovereignty on taking slaves into the territories. The Opening Ceremony<\/a> of the american revolution. The speakers david mccullough, michael quinn, and cookie roberts. It is my hope this beautiful new museum helps inspire you to become those active in citizens and is very great country. At eight on the presidency author catherine would talk about Florence Harding<\/a> and the new president she created. She had her kidneys operated on and have been in dire straits medically so she could relate to things they are going through. Out of veterans cause came the veterans. This was the first time the United States<\/a> had a barrel, what we call the virginia. For complete schedule go to cspan. Org. 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