 I'd like to bring up our introducer tonight, Laura Overdeck. Laura has been a friend of this museum and of me for many years, since before the museum existed, in fact. But she is better known at this point for being the founder of bedtime math. And for those of you who have young children or grandchildren or nieces or nephews or anybody in your world who you would like to instill a love of math in that person, you want to know about these books. And so I am delighted to welcome Laura Overdeck. Thanks so much, Cindy. It is great to be here. Great to see a packed house for what is going to be the most awesome presentation. So to warm us up for that, I thought I'd start with talking about what's wrong with America in case you got here and you didn't think anything was wrong. And what I specifically want to talk about is a specific fear, the fear of math. Now, you all love math, right? And even if you don't, you would never admit it in this room as you can keep that to yourself. But out there, lots of people really hate math. There are adults who are willing to say, I just can't do math, or I don't need math in my life. People are really nervous about it. They see an x and they just start sweating. And this is why restaurants now have to, on the receipt, print the suggested tip. Because people are not comfortable dividing by five or six to figure out what that number is. And this is a sad state of affairs. So how did we get here? There are a lot of theories about that. My theory is that today's adults grew up as kids who were never engaged in math, who never found it fun. They may not have really done math in the home in a playful way. So their only experience was school. In school, they probably had boring worksheets that didn't seem to have a lot of meaning. And worst of all, they may not have understood it. Because the way school has worked the last 100 years is that it moves along at a certain pace and all the kids in the class follow it. And if you hit a topic and you don't understand it, you don't have time to stop and dig into it and really understand it. The class is going to keep moving. So you have to memorize what you need to do. And what happens is you start having a pile of things that you've memorized that you sometimes carry the one and you're not sure why. Or when you add fractions, the bottoms have to match. But that isn't true when you multiply. Why not? There are reasons for these things. But if you have to keep memorizing and remembering all this stuff, it eventually turns into panic. And that's how you get the state of affairs today with adults, parents, as well as teachers often. I run into this all the time as the founder of bedtime math. Our mission is to help kids love math at a young age so that they grow up to love it as adults. We all know to read a bedtime story. That's a cherished tradition. It's warm and fuzzy. People read for pleasure. We don't have a lot of math for pleasure, right? How often do your people say that? So that time math is all about giving kids fun math at night. And what's interesting is we don't have a problem convincing the kids. The kids love it if they get a chance to try it. They love learning the math behind cool patterns or some weird animal with cool facts about it or food. In fact, right in this room, I led a big group discussion with kids about the 10-second rule. How long can food sit on the floor before you're going to die if you eat it? Kids had, this is a quantification problem, right? Kids had all kinds of ideas. Two seconds, 10 seconds, 12 seconds, one kid in the back yelled, infinity. And then I looked and it was my kid because we just ate off the floor in our house. But the thing is the kids get into it. They love it. It's the parents. It's the parents we're trying to convince. That's what's so difficult. And this is where we bring in Eugenia Cheng because she is a multifaceted, multi-dimensional person who is bringing it all together in such a fun way. She is a mathematician who lectures not only at the University of Sheffield but also at the School of the Art Institute in Chicago. That is a very interesting dual office to have. She is also an accomplished pianist and a passionate and talented cook. And what she is doing is tying in math with topics that people already love, like music and food. Her book, How to Bake Pie, is such an awesome bait and switch because every chapter begins with a recipe but then goes on to show how the thinking in that recipe ties in with mathematical principles. She, you will see this tonight, she makes a Battenberg cake where the squares, when you cut the cake and see the cross-section, the squares are different flavors that cannot touch the same flavor. But then it turns out there are all kinds of mathematical exercises that reflect the same pattern. In another chapter, she made a conference chocolate cake where she had surprise guests and had to whip together a chocolate cake with whatever was in the house, thankfully chocolate she had available. And her point there is you can't just memorize things. She had to be nimble on the fly in an unknown situation. Math is the same way, you can't memorize it. You have to feel it and understand the principles. Because when you are truly fluent in it, you are confident and confidence brings playfulness and when you then bring math and food together, playfulness brings yumminess and that is exactly what tonight's talk is about. So with that, I welcome Dr. Cheng. Thank you so much, Laura. And first of all, I would like to thank Cindy and Mo Math for inviting me here to talk to you today and it's so great to see a full house of people who want to come and hear about math in the evening. Now first, I would just like to say, are there any small children who are at the back and can't see enough? Is there some way we can bring, is there any way we can bring? If you'd like to come and sit on the floor at the front, then you're welcome to because I'd love you to be able to see small children. Oh, look, you can sit over here as well. Wonderful. So I'm going to talk about math and food and I love math and I love food and unfortunately most people love food more than they love math. See, some people love math more than food and I like to bring my love of food into math so that I can make a connection with everyone about math and I think that as Laura said, there are many things a little bit amiss with the way math is sometimes taught in school and I know there are probably some wonderful math teachers here in the audience because if you're a math teacher and you're here, it's probably because you're fantastic but unfortunately there are many other math classes and I know this because I teach art students now and my art students are almost all people who were turned off math in school and I talked to them, it's a bit like therapy, about their past math trauma and I talked to them about what it was that put them off math and over the course of my class in one semester I turn them all around so that they love math and that they think math is relevant to them because it's not just about numbers and it's not just about equations and formulae and it's not about memorizing multiplication tables and it's not really about memorizing anything and it's interesting because my art students tell me they got put off math because they had to memorize things whereas I loved math because I didn't have to memorize anything. I hate memorizing things except music but because I felt that if I understood it that was all I needed and if you do understand it that is all you need and so the math that I do with art students is the kind of math I think I can do with everybody. We build things, we create things, we explore just like you can in the kitchen and for me the most fun part of math started when I got to do research and I got to make my own things up because then you just invent a world, you invent a mathematical world and then you explore it just like when you don't have to follow someone else's recipe when you're cooking anymore it becomes more fun I think. You get to invent your own recipe and then nobody has to tell you whether it's right or wrong. You just decide whether you think it's delicious. I mean you might poison yourself but apart from that, apart from that it's just about whether you like it or not and in the end math is like that too. So I'm going to show you some of my favorite mathematical structures that you might not have thought of as being mathematical because math really is everywhere whether you see it or not and I think it's more fun when you do see it. So here are the things I'm going to talk about. I'm going to talk about Bach, juggling, hair, factors of 30 that is kind of mathematical, cake and custard. And so what is math? I have to tell you first of all what I think math is. Math is the study of how things work but it's not just any old study of how things work and it's not just the study of how any old things work. It's the study of how logical things work and not just any old study of how logical things work. It's the logical study of how logical things work. And the trouble with that is that nothing works logically. I don't work logically, you don't work logically, sorry but the computer definitely doesn't work logically. Nothing actually works logically and so in order to study anything logically we have to ignore the pesky details that prevent things that work logically and that takes us into the world of ideas. So we move from the real world into the abstract world of ideas where things really work logically and that move can seem scary because you don't really have your feet on the ground anymore, you're just working in the world of ideas but if you like ideas and you like logic it's a beautiful place to be. If you like winning arguments by yelling then you won't like that world because you can't win arguments by yelling. Unfortunately there are quite a lot of people who do like winning arguments by yelling but in this world you can only win arguments by making logical arguments and it's not even really about winning and losing, it's not about right and wrong, it's about exploring, understanding and thinking more clearly and wouldn't it be better, particularly in these days, it wouldn't it be better if more people were able to think more clearly. So first I'm going to talk about one of my favorite pieces of music which is a piece by Bach. Bach wrote many pieces of music and there was a lot of math in what he wrote so much that I could probably talk about that by itself for about three weeks so I won't do that but he wrote preludes and fugues in every key because a mathematical advance at the time meant that it became possible to tune instruments in every key. Previously you could only tune them in some keys and the other ones would all sound terrible but it was so exciting that you could actually play music in every key that Bach wrote a piece in every key and there are 12 keys on the piano so he wrote 12 but he wrote major and minor so that makes 24 and then he got really excited and did it all again so that makes 48 so he actually wrote 48 preludes and fugues and this one is from the second book in G minor it's one of my favorites if there were a piano I would play it but instead I have a video of me playing it this is my very tidy piano studio in Chicago and here is me slightly unkempt playing this piece hopefully we have sound polyphony which means that there are different voices of music that exist kind of independently and they wind their way around each other and you could in theory get some people to sing it so the first line in music is the one that the top line starts here and it goes ba dah and then the second line is this one that goes ba dah dah dah first studied this piece of music I got very confused I felt that I couldn't follow where the different lines of music were and so I decided to draw a picture of it and the picture came out like this this picture is a very extreme abstraction of the piece of music in which I've forgotten every detail apart from where each line of music is in relation to the others at any given moment and so this is the top line this black one is the top line but you can see that it comes down so it starts as a soprano and ends up as a tenor and this one starts as the alto and ends up as the bass and the bass line winds its way up and ends up as the alto and so now you can see that if you did get four people to try and sing it it would be a bit difficult because most people can't sing all those different parts I have one friend who can sing soprano, alto, tenor and bass but that's only one and I also began to understand why I didn't understand and when you understand why you don't understand that's the first step to understanding something and I realised it was because the lines of music cross over and don't come back usually in polyphony they cross very briefly and come back again but because they kind of creep up the back I lost them once I drew this picture I could understand the structure, the abstract structure better and then I felt I could play the piece better and this is the thing that you don't have to see this picture in order to listen to the piece and enjoy the piece but I feel I can play it better because I understand this picture and it's the same with math you don't have to understand the math of your cell phone to use your cell phone and you don't have to understand the math of cars to drive a car you don't have to understand where the structural walls are in this building in order to use this building it's a good thing someone knows where they are and the math is everywhere whether you know or not people sometimes say to me you know what I get on fine without math and I say yeah but I get on better so if you understand how things work you can use them better but best of all you can invent your own things you can make things better you can improve them you can break them when they're fit well you can break them you can break them you can also fix them when you've broken them and so this to me is what math is about and this abstraction means that lots of different things become the same that were previously not the same so for example if you say if I have one apple and another apple that makes two apples and if I have one banana and another banana that makes two bananas and if I have one monkey and another monkey that makes two monkeys and you go oh something's going on here if I have one thing and another thing that makes two things so I might as well just call that one add one equals two and that right there is an abstraction from things to numbers numbers are already an abstraction that you've all got used to and you've forgotten doing it if you help small children maybe you have some younger brothers and sisters if you help them learn how to count they're confused at first and making that leap from objects to numbers is a big leap and the next leap that usually happens is from numbers to letters and people often say to me I was alright with math until all those numbers became X's and Y's and then sometimes people get on fine with the X's and Y's but then they get stuck when it's calculus and sometimes people hit their ceiling of abstraction in the middle of their PhD and that's really tricky so people hit that ceiling at all sorts of different I haven't hit mine yet I don't know if I have one but anyway one of the reasons I love abstraction is because it makes lots of things the same so you can study them all at the same time and that's good because I'm very lazy and math actually comes from mathematicians being so lazy that they don't want to do the same thing over and over again so they always make a theory that does it for them and math is there to make our lives easier actually you might think it's there to torture you in school but it's actually there to make our lives easier so I'm going to show you some things that become the same when we talk about when we abstract from them now these things are called braids in mathematics and actually I drew this picture when I was still an undergraduate and then when I started my PhD these braids appeared in the research that I was reading and I didn't know that was going to happen and they're also related to juggling I used to say I can't juggle and then someone said to me you know that's like when people say I can't do math so I stopped saying I can't juggle instead I said I'm practicing I'm getting better at it I'm still not really good enough to do it in front of an audience so I wonder if someone would like to do a juggling demonstration for me is there someone who can juggle hooray give a big round of applause what's your name? Ariel Ariel, wonderful here are some non-juggling balls but they're fruit they're delicious fruit if you drop the apple it will smush he's not going to drop anything that's it this is just going to be a juggling show now now would you please walk across the first stage and keep doing just the normal juggling pattern and see can you imagine if we had a long exposure camera what would happen if we took a long exposure picture so that the fruit makes paths through the air could you just walk across can you see the pattern? would they be in a coil? here is an actual long exposure picture I made thank you very much a big round of applause that was wonderful thank you so I made this picture with illuminated juggling balls I made this picture at a school in Chicago when I tried juggling it was so wonky that it didn't make a very good picture so we found a third grader who did this juggling for me here's one I made on the computer and so if you can see that we have what we have here is a red apple we have a green orange and a blue lemon so the green apple goes across like this and at the same time the blue lemon crosses over this way and while the blue lemon is going the red apple goes that way and it makes this picture like this and it is really a braid and if I turn it this way up then you can see it's the same as the braid in my hair and this is another kind of braid that we study in mathematics I once tried to braid my hair in the bark braid you can see that doesn't work at all because you would have to have a band here a band here, a band here it just doesn't stay in place at all whereas we all know everyone who's ever braided hair knows that when you do a normal braid you only need one band at the end and if I take this band out and then I pretend I'm in a shampoo commercial which I actually I always wanted to be in a shampoo commercial when I was little there are several things I wanted to be in a shampoo commercial and on a cooking show anyway I actually always wanted to be a mathematician but that's a whole nother story so this braid a mathematician's study how tangled up braids are and we study this mathematically it's a part of a field called topology which is about shape and the fact that this one is tangled up enough to stay in place has a mathematical explanation whereas this one, the green band is floating around and the black strand is floating around whereas if you look at just the red and the blue ones they're interlocking with each other so they won't move now this is something that mathematicians study because it's interesting and a lot of the things we do we just study because they're interesting not because we see a particular use for it but however this turns out to be useful because potentially it might be related to early diagnosis of Alzheimer's disease because they think it might be something to do with the way brain cells get tangled and if they're mutated and tangled the wrong way we need to be able to detect when the tangle has changed mathematically so it could be useful but that's not the first reason we studied it and math often happens like that sometimes it takes 2,000 years or 3,000 years for a piece of math to become actually useful and so often the use is more just in how it helps us to think and it's just like I think math has this unfortunate burden of being there's a demand on it that it should be directly useful and I think this is like trying to get a young person to be interested in something by saying it's useful I think it's a bit futile it's like introducing someone and saying I'd like you to meet my friend they're very useful so I think that interesting is a better thing to say about math and friends I couldn't make this braid in my hair so I made it into a pie instead this is my Bach pie it is banana and chocolate so it's a Bach pie and we're now going to think about some factors of 30 and we have some activities that are going to come round for the factors of 30 but first of all let's just think about what the factors of 30 are can we remember? yeah, what... start at the lowest what's the small... one two five six ten fifteen and thirty, very good that's not very interesting it's a bunch of numbers in a straight line it's kind of interesting but it could be more interesting see, we live in a three-dimensional world so unfortunately we write on two-dimensional pieces of paper in one-dimensional straight lines and so we squash everything into one-dimensional straight lines when really they have natural geometry that we've squashed and this I say is why I don't like to tidy up the papers on my desk because they have natural geometry where they are so let's discover the natural geometry of this situation where we're going to see which numbers are factors of each other and then we can draw a kind of family tree because family trees are there to show relationships between things so thirty is kind of the great-grandparent here and then we have six, ten and fifteen that go into thirty five goes into ten and fifteen two goes into six and ten three goes into six and fifteen and one goes into two, three and five so now we see that it's the corners of a cube which I think is much more interesting than numbers in a straight line and it's a cube a different way up from the way we usually see it because usually we think of cubes like this on a face but this is a cube on its corner and so the six, the ten and the fifteen are the three corners at the top and if I rotate the cube by its corners you can see those three corners rotating between each other and it's related to the little brain teaser that says if I suspend a cube by one corner and drop it halfway in water what shape does the cube make on the surface of the water? and they're very good you would write the first time the answer is a hexagon and so these little card things like we've handed out which the museum has beautifully laser cut offer us all to make models of a cube cut in half along the hexagon face, yeah do you have a question? mhm, yeah so we're going to make them now so everybody should have these templates, the idea here is that if you can see there's some little little lines which are laser cut lines for you to fold along so if you just fold along each of those lines fold them all in the same direction then you should find that the numbers match up so all the ones will match up and just fold them so that they make right angles and then it should turn into this thing and you can tape it together and then you'll see the hexagon face that appears so we need, is there a yellow, are there any more yellow ones? well what happened, if we could just have a few yeah, yeah and now you can build different things you can turn them backwards as well and turn them into other things I'm going to say 23, 27 if you get one point 0, 0, 0, 0, 0, 0, 0 when it's done the first thing you can do is you can hold it together to make a cube that divides into two equal halves in the non-obvious way but then you can also join up with your friends and if you turn them backwards you can build all sorts of different things with them so if I put two together backwards then they make this and then if you can get eight then you put them, if you put eight together then they make something which is actually a truncated octahedron which is quite fun you can also do this and if you put eight together this way then it makes a cube with something cut out of the inside so these things make lots of different things but if you just put them together in the ordinary cube then if you position it correctly then you can see which numbers are factors of each other and then you see that one and the prime factors are in one half and the non-prime factors are on the other half and I think that's a much more interesting structure of numbers than just a bunch of numbers in a straight line you can connect them anyhow you want and you can take them home and you can keep doing it at home so the other thing that we can do with a cube relates it back to the braids that we had I can wrap up this cube with ribbon as if it were a present and so here I'm going to use red and blue ribbons and then those red and blue ribbons are going to carry on down the sides of the cube and now I need a third colour it's going to come across here so that makes a green going across like that and if I wrap up the other present with the same colours starting in the same places so the red will start here and the blue here and the green here then it looks like this and now maybe you can see that if you shuffle the red and the green crossing down a little bit and the green ribbon up this turns into that so mathematically those are the same braid and this is something that has a fancy name in math it's called the third Rydermeister move we like to give things fancy names in math because it makes us feel clever but actually it's just to help us remember what things are and lots of mathematicians, some mathematicians unfortunately like talking in the strange words specifically so that people won't understand because it makes them feel clever I don't believe in that I believe in bringing everybody in this is something else that I saw during my PhD and I realised that I could try it on my Bach braid so this is the one where you start with a cube and you put the ribbon on it you wrap it up and I thought well how about I start with the Bach braid and I make a present and I reverse engineer the present to be the right shape to fit under that braid and I thought about it and I discovered that it's a cube and when I realised it was also a cube I did exactly what I do whenever I discover something sort of prove a mathematical theorem I go oh my goodness that's fantastic and then two seconds later I go oh I'm probably wrong and then two seconds later I go maybe it's trivial and then two seconds later I think I bet everyone in the world knows this apart from me that is honestly the process that I go through every time I prove a theorem so like a good mathematician I checked it I thought I wonder if every braid makes a cube and I realised no it needs six faces to be a cube so it needs six crossings so I thought maybe every six crossing braid is a cube but then I realised that's not true either because you could just have two strands and go cross cross cross cross cross and that wouldn't be a cube so I don't know why this is a cube I phoned a friend my friend who actually sings soprano alto ten a bass and I said oh my goodness and he said I think it shows that Bach was communing with the spirit of the platonic solids and I don't think that's what was going on I don't know why this is happening and I don't know if it's useful but I think it's cool and I think that's enough that's enough for me anyway and there's a lot of math I just think it's really cool and that's why I'm showing it to you the next thing that I think is cool is cake Battenberg cake especially it's one of my favourite kinds of cake of all cooking I most like making desserts because they are nutritionally valueless anything that has actual nutritional value I'd rather leave to someone else this is only there to make me happy and isn't that nice Battenberg cake is a very English kind of cake it's supposedly related to the crest of the Battenbergs or something but the main point is that you have two colours of cake and you definitely don't want the same colour to touch itself because that would be terrible so this is how it looks and it's actually the same as a mathematical structure we can do multiplication of 1 and negative 1 and here we're going to put 1 times 1 which is 1 usually and here we're going to put 1 times negative 1 which is negative 1 and here we're going to put negative 1 times negative 1 which is 1 so it's a Battenberg cake and we can try this again by adding even and odd numbers here we're going to put even number plus an even number even and here if you add an even number and an odd number you get odd and if you add an odd number and an odd number you get even so it's a Battenberg cake and we can now try this with multiplication of real and imaginary numbers now you might not know you might not remember what real and imaginary numbers are but imaginary numbers are that thing that happens when they tell you at school you can't take the square root of a negative number and then the next year they tell you now we're going to take the square root of a negative number so imaginary numbers are what happens when you take the square root of a negative number now I'm not it doesn't matter if you know what they are I'm going to tell you that they make a Battenberg cake so a real number times a real number must be real and a real number times an imaginary number must be imaginary and an imaginary number times an imaginary number must be real because I told you it's a Battenberg cake and now we can extend this and we can talk about bed flipping mattress flipping is what you're supposed to do every season to stop your bat bottom from digging into the same part of the mattress all year I never do it because I'm much too lazy but if I were going to do it I would definitely write letters on my mattress and this is why I've given you something with ABCD on it to remember where I am now this is the trouble with supposedly real life math problems usually they're nothing to do with real life at all like they say things like you have 87 wild horses and 15 of them escape because yes I definitely have 87 wild horses and my friends often send me the homework questions that their children have been given when they're really ridiculous and my favourite one to date has been you have one metre of string you use 30 centimetres of it to tie up Tommy you you use you did this one you have you use 35% of it to tie up Adam how much do you have left to tie up Sam and apart from the question of why we're tying people up 30 centimetres is not very long to tie a piece of string to tie someone up with so anyway what we can do is we can we can make we can make a little table of what happens to our mattress and here is where you do nothing and then you do nothing again because you're me and you're too lazy to flip your mattress so you always stay if you start in the A position you always stay in the A position here is where you do nothing and then you rotate so this is what I'm calling rotate and that gets us into the B position so if we start in the A position and we rotate we get to be rotate and then here we're going to start in the A position and then we're going to flip over sideways and that takes us to C and if we start in the A position and flop over that's what I'm calling flop that takes us to the D position now going down here we're going to do the same move and then nothing afterwards and that gives the same answer because if we do nothing it doesn't matter when we do nothing I do it all the time frankly but this gives us the same answer so let's try doing a rotate and then a rotate so if we start in the A position we rotate and then we rotate again it gets us back to A and here we can try flipping and flipping again so we start in the A position we flip over sideways and then we flip over sideways again when we get back into the A position and here we can try flopping so we flop over and then we flop over again and this gets us to the A position again so that tells us if we do the same move every season we're always going to miss things out because we keep coming back to A so we can now try mixing them up we can try doing a rotation and then a flip so we rotate and then flip over sideways that takes us to the D position and here we can try rotating and then flop so start back in the A position so we rotate and then flop that takes us to the C position and here we're going to flip and then rotate so start in the A position we flip over sideways and then we rotate D and here we're going to flip and then flop and this is the whole point I get to say flip flop B very good now maybe you can guess what's going to be here C very good and what do you think guess what's going to be there B very good and now because we're good mathematicians not people on the internet we could check that it's really true so we can check this flop and then rotate so we can flop and rotate and see that we really do get that and we can check the other one as well map is often about spotting patterns and guessing it's not all about knowing it's about guessing the right answer and then checking using logic now the real question here is how many battenberg cakes can we see so how many battenberg cakes can you see in that picture there's these ones there's one in the middle and there's the there's the big one where it's a cake battenberg cake each of whose cakes was already a battenberg cake here's one I made and here's where I put the letters on to show you what it is I call this the iterated battenberg cake because I have iterated the process of battenbergification and battenbergification is basically like painting by numbers so if I tell you that we are battenberg simplifying all I have to do is tell you what goes in position one and what goes in position two and you can make the picture so it can be yellow and pink it could be chocolate and more chocolate because that's always good it could be a bunny and another kind of bunny or it could be one kind of battenberg cake and another kind of battenberg cake and that's why it's the iterated battenberg cake if you try to make this cake you have to cut off so much cake to make the squares perfect that you end up with more leftover cake than cake which is good for your friends I can make an even more abstract representation of this because all I have to do is tell you what goes in position one and two so I might as well just draw it like that so it could be yellow and pink it could be light brown and dark brown it could be bunny one and bunny two it could be cake one and cake two now what is cake one? cake one is one of those and what is cake two? cake two is one of these now I might as well put the whole picture for cake one in this one spot and I might as well put the whole picture for cake two in the two spot and now I've made an extremely abstract representation of the battenberg cake where what I've told you is how these four ingredients get put together to make the final result and this is what mathematicians study and they're called trees because they look a bit like trees and they show us how we stick things together now here are two trees showing how to stick three things together in two different ways this one says put the first two things together and then the third one afterwards this one says put the second two things together and the first one afterwards and we can ask if we get the same answer so we can try this with adding numbers together if we add four and four we get eight and if we add two on afterwards we get ten and we usually because we squash things into straight lines we usually represent this using parentheses because we have to put in a straight line instead of the tree the other way round says put add four and two first which makes six and then if we add four afterwards it it makes ten and this way round the parentheses go over here and you might remember this as the associative law for addition but really it's about moving branches of trees around and if we didn't have to squash everything into a straight line it would have been represented by these tree pictures instead now the thing is that numbers behave like this but when we study more subtle things than numbers they don't necessarily behave like this for example if we add egg yolks and sugar and then we add milk and heat it up we get custard the other way round says mix the sugar and the milk first and then mix the egg yolks and we get we do not get custard so these two things are not the same however if you're making cake it doesn't really matter so much but the cake mix the cake recipe I've always used says first mix the sugar and the butter then mix the eggs in and finally fold in the flour and that makes cake but actually it turns out especially if you have an electric mixer basically you can just chuck everything in a bowl and go and so it's fine and that's what that tree looks like now if you actually try all the possibilities and I brought all the cake ingredients into my art class and we mixed cake in lots of different ways and here are all the different ways you can mix cake so each of these arrows represents moving one branch across just like this so here I've moved the butter branch across so this new recipe says mix the butter and the eggs which by the way looks really disgusting and then mix the sugar and finally mix the flour afterwards but actually it turns out that if you beat it hard enough it kind of turns out alright now this one at the bottom is interesting because it says mix the sugar and the butter but also mix the eggs and the flour so you need two bowls and if you're me and you're lazy you don't want to wash an extra bowl so that really makes a difference now this one over here says do the whole thing backwards mix the eggs and the flour which is a very odd thing to do and then the butter and then the sugar but again if you beat hard enough it turns out alright however we do not have ovens at art school unfortunately so we couldn't actually bake it to see what happened we just looked at it and said looks kind of the same however I went to speak at the culinary institute of America recently and they do have ovens and so we baked it to see what would really happen and what happened was it was all like cake it's just that some of it rose more than others so the normal method was definitely the easiest and it rose a lot but actually I think it was this one and this one that both rose quite well whereas the top right one and the bottom one were really flat and so in the end they still looked like cake and they still tasted like cake and actually I quite like dense cake so it depends what you like about your cake it also depends whether you're the person making it or not it depends if you care that halfway through your cake process your batter looks like vomit and when you're doing higher level math you get to decide a lot more about what counts as the same or not when you're just doing numbers things are just equal or not when you're studying more subtle objects there are lots of different ways in which things can be the same just like if you offer a child two pieces of chocolate cake they might really care which one they get because one of them will definitely be better than the other whereas as adults we sort of get used to the fact that it's okay that most of the pieces of chocolate cake are more or less the same so there's a lot more decisions and opinions you get to have in math and if you make a different decision you just go into a different world it's like a choose your own adventure thing probably don't have those anymore in the world of modern technology but I'm showing my age I used to have choose your own adventure books where you choose what you want to happen next and there's something fun about making those choices and math is really about making those choices unfortunately too often when we teach it at school we don't let anyone make any choices and I think if we let everyone make some more choices then you have more ownership over what you're doing so I want to finally end by saying that category theory is what my field of research is and there's actually been quite a lot of category theory this is something that comes up only if you're doing a PhD in category theory really are you a category theorist? Oh hi, so this is actually something that's very important in my research it's called the McLean Pentagon it's about it's about cake really and if you want to make chocolate cake that also comes up because I do higher dimensional research if you want to make chocolate cake you're going to have one more ingredient so all your trees will have one more leaf and then the shape that you get if you draw all those is one has one more dimension and here it is someone 3D printed it for me and there's a picture in the book of a thing you can cut out and fold up to make it so here it is it has 6 pentagons and 3 squares and we build this in my classes with the art students although there are art students many of them can't multiply they definitely can't divide and yet they're doing this thing that I only did when I got to my PhD which is in category theory which is the mathematics of mathematics I say and if mathematics is the logical study of how logical things work and category theory is the mathematics of mathematics then if we unravel that category theory is the logical study of the logical study of how logical things work and I believe that this is all things that we can show everybody without you having to pass endless tests and memorize endless formulae because it's really about creating worlds and making things delicious so thank you very much for coming alright so we have some I think some time for questions in any categories it doesn't matter so if you could raise your hand I will bring a microphone to you what was the change in instrument design which allowed the notes, individual notes to be tuned independently as opposed to independently oh it wasn't exactly that they were being tuned independently was that they used to tune using harmonics and if you tune using harmonics then all the strings have ratios that are rational numbers so they're two thirds and a half so that the harmonics match up with each other and in order to tune all the notes evenly so when you do that C and G are very beautiful for example and F but then F sharp sounds terrible and so to do it evenly you have to make the square root of 12 so that the half step has to be a ratio of the square root of 12 because wait I didn't mean that, the 12th root of 2 because when you multiply it 12 times you get to an octave and that's a difficult thing to figure out if you don't have some advanced mathematics so actually there's some story allegedly someone in China figured this out on a huge abacus a kind of 150 column abacus hundreds of years ago I'm not sure if that's actually true, I once read it on the internet so so it's probably false but then they figured out how to make it once all the intervals are even by the 12th root of 2 then they're all sort of equally out of tune none of them is actually the octaves are all in tune but the fifths are slightly off and so the everything is slightly off instead of a few things being very off, it's kind of like the bump in the carpet, if you have a bump in the carpet you can push it over there or you can spread it out evenly everywhere and so now we spread it out evenly everywhere so everything sounds more or less okay that's what happened are there any more questions? Yes when you encountered Gerdl's incompleteness theorem did it cause you any consternation? Gerdl's incompleteness theorem says that any logical system is either feeble or it necessarily has questions that can't be answered the reason is because if you're powerful enough to express self-reference then you can use self-reference to create weird situations like this sentence is false or the next sentence is true, the previous sentence is false and then you can get yourself into weird loops, I love those loops, it causes me no consternation whatsoever I don't believe that mathematics is there to answer all questions I believe that mathematics is there to help us understand what is at the logical core of the world I have this image that there's logical stuff that's in the middle of everything and there's illogical stuff that's everywhere else and I don't think that's bad because there are some things that we can't use logic to understand either because logic is too slow or because we simply don't have enough information or because there are just too many variables or because it involves things like human emotions which do not behave according to logic and that's wonderful and without those non-logical things I think there wouldn't be the beauty of music and poetry and love and to me the most exciting part is where the interface is just between the things that we can explain by logic and the things we can't so in music I love trying to understand what is making me so excited about the music and I'll analyze it and analyze it and find everything I can and just beyond all the things I can explain there'll be one last thing which I can't explain and to me that's where the beauty is just on the boundary and I have this image that logic is the sphere at the center of everything that we understand and that we should try and put as many things as possible in that sphere and as we do the sphere of logic grows so the surface grows and that surface is where beauty is the interface between logic and illogic so the more we understand with logic the more access we have to the things that are beautiful it's like a rising cake yes alright so unless someone is desperate for a question I think that it's well past some folks bedtimes here fine let's give Eugenia another round of applause thank you so much