IC16: How To Bake Pi

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IC16: How To Bake Pi
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Mathematics can be tasty! It’s a way of thinking, and not just about numbers. Through unexpectedly connected examples from music, juggling, and baking, I will show that maths can be made fun and intriguing for all, through hands-on activities, examples that everyone can relate to, and funny stories. I‘ll present surprisingly high-level mathematics, including some advanced abstract algebra usually only seen by math majors and graduate students. There will be a distinct emphasis on edible examples.
Point (geometry) Mathematics Different (Kate Ryan album) Algebraic structure Mass Nichtlineares Gleichungssystem Mereology Connected space
Mathematics Process (computing) Observational study Logic Different (Kate Ryan album) Multiplication sign Order (biology) Mass Numerical analysis Condition number
Different (Kate Ryan album) Algebraic structure Mass Musical ensemble Mereology Resultant
Group action Building Process (computing) Theory of relativity Observational study Block (periodic table) Multiplication sign Independence (probability theory) Algebraic structure Line (geometry) Mass Student's t-test Limit (category theory) Mereology Mathematics Goodness of fit Order (biology) Chromosomal crossover Musical ensemble
Axiom of choice Greatest element State of matter Multiplication sign 1 (number) Mereology Mathematical structure Group representation Mathematics Roundness (object) Many-sorted logic Different (Kate Ryan album) Oval Arrow of time Körper <Algebra> Algebra Position operator Rotation Process (computing) Block (periodic table) Connected space Category of being Angle Order (biology) Right angle Lipschitz-Stetigkeit Mathematician Multiplication table Imaginary number Point (geometry) Divisor Observational study Parity (mathematics) Real number Student's t-test Mass Rule of inference Graph coloring Event horizon Theory Pentagon Causality Energy level Nichtlineares Gleichungssystem Curve fitting Condition number Platonic solid Addition Matching (graph theory) Weight Algebraic structure Line (geometry) Braid Cartesian coordinate system Numerical analysis Logic Network topology Cube Vertex (graph theory) Faktorenanalyse Mathematical object Iteration Musical ensemble Family
Greatest element Group action Block (periodic table) Multiplication sign Polynomring Algebraic structure Routing Student's t-test Mass Proof theory Mathematics Root Different (Kate Ryan album) Order (biology) Number theory Integer Right angle
Randomization Group action Sine Multiplication sign Mass Student's t-test Inverse element Mereology Dimensional analysis Theory Element (mathematics) Pentagon Mathematics Conjugacy class Many-sorted logic Different (Kate Ryan album) Square number Extension (kinesiology) Social class Dependent and independent variables Shift operator Cartesian coordinate system Category of being Network topology Cycle (graph theory) Object (grammar)
but even in the air and and thank you thank you very much with but so I'd like to thank the organisers for inviting me to come in and talk to you about abstract mathematics and my experience of making abstract mathematics more palatable to people who use she afraid of math and I understand why people are afraid of math because the the weights toward because of their experiences when they're young and I also found on in Europe I should say that I have also I also found math rather boring in school but I always knew there was something more beautiful out there waiting for me partly because my mother showed me the most beautiful and pull parts of mass so I believed that it was there the heart of trained myself this enough the so I'm going to show you some of my favorite abstract structures that appear all around us because I think that math is not about doing calculations or about doing equations or even about solving problems it's about as interested in in making connections between things and the point about abstraction is that if you lead into a slightly more abstract wealth than your further away from normal life but you make connections between lots of things that previously seemed to be unrelated and that a whole different kind of light on things so 1st of all what I tend to do is and this is why I wrote this book I want to persuade people that map is not the thing that they necessarily thought it was so
I think that mass is the study of how things work and it's not the study of how many things work it's the study of how logical things work and it's not any old study of how logical things work it's the logical study of how logical things were and the trouble with that is that nothing actually don't behave logic I don't behave logically you behave logically the computer doesn't behave logically nothing behaves logically so in order to get anything to behave logically you have to forget the pesky details that prevent things from behaving illogically and that the process of abstraction where you forget the details and I love forget some details and in the process of getting those details you do things like well what went on another on makes 2 honors and 1 monkey and the monkey makes 2 monkeys and you know that really matter that that was a ban on all that it was a monkey 1 thing and another thing makes 2 things and so you might well call that 1 and 1 equals 2 and that's already is abstraction everyone performs when they 1st start thinking about numbers and usually every time you go through another abstraction some people that get full and a lot of people said no as by
with mass until numbers turned into X and Y and that's an abstraction and the thing is that that usually people often shown abstractness is just some weird thing that you're doing for no particular reason rather than an amazing way of making things easier and the secret is that conditions are mostly very lazy I'm definitely lazy and the reason I like is it means I can think about a lot of different things at the same time which saves
the effort rather than having to think about and lots of different things so I would like to show you some of my favorite abstract structures and shows some perhaps unexpected ways in which they're related so I'm going to start by playing you 1 of my favorite pieces of music which is the 2nd g minus the boss prelude in G minor from the 2nd book of proteins and future there is a huge quantity of mass in and I'm not going to say any of that I'm going to say this about what the at the question of the moon and the US and a 2 room and you will be in the room and the and the and the and the way we do in
the room and the new and the old of the a new world and you can do and in the in the in the room and you know you and you can the thing that you have to do with it and as a result of the of the of the of the of the of the and the and the and the and the other part of the and the and the the the the the the the
the the the the the the a lot of people in the world and of the few you don't have to be able to play the piano to appreciate a piece of music and you don't have to be able to cook in order to be able to appreciate food and I believe you don't have to do that in order to be of appreciating that and IT t to the school of the art institute in Chicago where I teach our students and my aim is to get appreciate why there and what it's doing even if they're not going to go out into the world and use mass in their job they already PhD's in that they made engaged in something else but usually people but they all at some point in that they get reach their limits in mass testament Anderson warning NORAD and yet there's so much that to be appreciated now in the block wrote it as politically like a lot of the pieces roads which means that our full independent lines of music that wide and their way around each other and the first 1 starts at the top like this on our group and so you can play each 1 independently and when I 1st study this piece I found it very confusing and I felt like I couldn't follow where the parts were going so I drew a picture of it and the picture came out like this and this picture is a very extreme abstracts of the piece of music in which I have forgotten almost every detail about it apart from where every line of music is in relation to the other lines of music so this is the abstract structure of the peace and exhibiting this abstract structure made me understand why it I didn't understand it because the crossover each other and they don't come back again so this this black 1 which starts the top is the soprano but ends up as the tenor and the base starts as a base that creeps up and ends up as the outer and so be very difficult to get for people to sing this because they will have to be of the same old parts which people can't you do the said finding the abstract structure the piece I believe helps me play better even though in order to listen to the piece you don't have to see that structure in order to listen and enjoy the music just like it's a good thing someone knows whether structural walls in this building on but we don't have to know whether structural walls or in order to use the building and that's what's going on with math all the time it's that whether you know it's there or not and the more that you understand about how things well 1st of all isn't it nice to understand things better just that is secondly enables you to use it better make it work better and then maybe change it and invent something new for yourself and I believe in presenting
masses something where everybody can have a go invented something new just like with cooking which is why I talk about the a lot in my book because really matters about taking a bunch of ingredients which happened to be ideas doing something with them and seeing what happens and too often we impose on students the notion that they have to get the right answer instead of getting an answer they feel like as long as they have a big
logic and then deciding for themselves whether they like it or not so I I think with cooking it's the same thing if you're at a cost then you might be afraid that you have to make the right thing but when you cut yourself it doesn't matter all that matters is whether you like it or not and hopefully that you don't poison yourself and I so moving on to juggling and very bad at juggling I'm not supposed to say that this is someone pointed out to me that's like saying I'm very bad enough of however I'm not very good at juggling and Brody go wrong if I try and do it in front of people so can only a job wonderful would you like to come into juggling demonstration about the same the juggling and holding a microphone at the same time can you javelin while walking across the stage wonderful so if you imagine that there's a long exposure camera as he walks across the stage the bulls trace a path through the air and the path looks like that so you start with state the green and the red on 1 hand the gringos across yeah and then they get themselves looped up thank you very much that's a preview for Henry who the the lakes and it's a bit like this picture is a bit like that picture musical braids in maths and we study them as mathematical objects a bit like the not that of someone whose name of from fortunately was talking about the other day on and it's also just exactly if you turn it this way it's just like the brain in my head and mathematicians study how you up the parts of the brain also this you can tell is totally talkers only need 1 bound to hold it together if I take that out the 10 diminish then you see it completely falls apart whereas this parade I once tried splits in my head that you can probably see why that's not going to work at all because it's not only top is green 1 is sort of free floating so to hold it together you would need a band here a band here and that band they're not very practical so instead I made in supply and this this is my boss apply it's banana and chocolate which is what sparked by the so called people I started I drew this picture before I started my PhD in category theory and during my PhD I discovered that breaks was central to the study that I had happened to the chosen and realized I had drawn 1 years before without even realizing it so it was an unexpected connection and here's another unexpected connections with factors of the so the factors of that the their numbers that people think of numbers as being maths and maths is being numbers matter so much more than numbers the fact that the all 1 and to 3 5 6 10 15 and 30 that's not very interesting a bunch of numbers in a straight line nor interesting on forging that's what we throw children school hold on numbers in a straight line however if we also exhibit how the numbers of factors of each other we can draw a kind of family tree of them and so we get 30 that's is kind of great grandparents at the top and then the next 10 and 15 go straight into 30 with nothing being in between 5 those into 10 and 15 to those into 6 and 10 the reader into 6 and 15 and 1 goes into 2 3 and 5 and now we see it's the vertices of a cube on its corner and it's quite an interesting psychological factors small children don't think a cubist in the Cuban missile sitting straight on its face but anyway this is a Cuban 1 it's called a which is not the usual way that we think of caves but I think that's much more interesting than a bunch of numbers in a straight line and kids come up a lot in my research as well in fact we can take the keep apart and then wrap it up with ribbon because why not sometimes dishes do things just because it seems like a fun idea and some useful application might happen later medieval happened 200 years later made it happen 2000 years later like the i costing Heejin that is 2 thousand years old in 2 thousand years later we discovered that by Costa he dreams all what the shape economy my sentence work many viruses or eicosa Heaton shapes and that was the 1st time that of naturally occurring Michael's Eden was found 2 thousand years after that the Platonic solids were sort of abstractly anywhere that's a wrap this up with ribbons that crossover like this my rule cause I can make up any rules I want my rule is that the right hand ribbon is going to go on top of the left and 1 is coming in from the left so I need another color to come down here as the green like my juggling balls and on this I'm going to start with the same red colors coming out to the same places that's going to look like that and then maybe you can see that if you shuffle this green ribbon up and shuffle the blue and red crossing down this braid crossing will turn into evaporated crossing and so that's what mathematicians consider to be the same because if you can shuffle them around you're really just looking at it from a different angle and this is called the braid crossing thing is called you may have heard of it the 3rd Reidemeister move and it's it's 1 of the moves that you can use to see whether to not really the same not and it turns out that this might be related to early diagnosis of Alzheimer's but but that's not why mathematicians started studying in the 1st place the cute version is something out of high-dimensional algebra which is what my field of research is and it's called something else but the fact that the 2 are related is a very profound a structural relationship between different fields of maps and he's after doing this I suddenly thought what would I do that to my boss break what will happen so here is my boss operates so if you imagine that your wrapping up the present with this you do it's a kind of reverse engineering what shape of present which you have to have in order to wrap up a present and get these crossings of ribbons and I stare at it for a while and you know what it's a q and I have no idea why and I told a friend of mine and he said 0 I think it's because block was communion with the platonic solids and I don't think it was about I I don't know why I don't think there has to be a reason for it but I think it's pretty cool anyway and a lot of math is just back it's cool MAD 2 thousand years later some will do something useful with it but in the meantime why do we need to burden ourselves with it being useful all the time call this is kind of its own and sometimes and now we move on to take which is kind of not very useful which is why I like it I really like desserts because they're not there to be nutritional they have very little nutritional value there the delicious and cause joy that's its usefulness the pattern but take is 1 of my favorite kind of cake the it's very English cake it's something to do with the crest of the pattern bugs or something but the main point is that you have to colors of cake and you definitely don't want the same color to touch the same of a color because that would be terrible so this is actually a mathematical structure you may recognize it for example if you do multiplication table of 1 and minus 1 well here you put 1 times 1 which is 1 and usually paper 1 times minus 1 which is minus 1 EPA put minus 1 times minus 1 which is 1 so it's about the case and you can also that would even or odd numbers if you add them up not if you multiply them and even number plus an even number is even and even number plus and order number is all and any and all and all of them the plus a node number is even so it's about the cake and since we're out matter a conference like that that you can do this with multiplying real and imaginary numbers and you also get the Battenberg cake does that's nice you can relate this to bed flipping Which is why I've handed out little cards with a species and these on them but I want my hands so I should do this is the best of flip your mattress every season to stop your from bottom from weighing into the same part the matches all year so obviously I don't do this because I'm much too lazy and that sounds like hard work but if you were going to then obviously you'd write a B C and D on your matches this then we will write letters on on mattresses that this is 1 of those things which can seem very spurious about claiming that matters related to real life I'm not pretending this is real life but it's kind of fun anyway so if you flip them all things you can do you can do nothing which is which is what you do if you me so you nothing all year then you stain yea positional time assuming you've started in a position you could also during rotation which I'm saying is like that which takes you into the B position if you start in the a position again you can whip sideways which takes into the sea position and if you Thorpe and clean up for this way then you get into the D position the if you do each of those things and then do
nothing afterwards you get the same answer because it doesn't really matter when you do nothing you can do anything you want but if you do a rotation followed by rotation so if you do a rotation and they need another rotation you get back in a position if you do have layup and then another slip you also get back in the position and of course if you do a fall and then another flop you also get back in the a position which shows you can't keep doing the same thing every season could you always keep an imaginary position and you always escaped some places so now we have to combine something will have to you but we can so we can try doing a rotation and then a flip which takes us to the we can try doing a rotation and then a fall the viewer rotation and then the fall that takes us to see and if we do a lit and then a rotation of the lips and then a rotation we get the and then maybe you start seeing a pattern which is how a lot of math research goes you know you you want to save yourself some works the spot pattern and then try and justify off to the event which is often easier than justifying it before spotting the so what you think is here the be embody thing goes here see and embody thing goes here be so if we organize conditions we should protect that but unintelligible works and then the question is how many Battenberg cake can you see in the picture so there's a these ones there's also there's 1 there's 1 in the middle but there's my favorite 1 which is the fact that the whole thing is a giant pattern but K each of whose cakes was already a Battenberg cake so here is that it's an iterated Battenberg cake and I call it the iterated Battenberg cake is what I've done is I've iterated the process of pattern but ification Battenberg ification 7 and taken great than a pain like numbers so all I have to do is tell you what's going to go on position 1 and in position 2 so they could be yellow and paying all it could be chocolate and more chocolate it could be a bunny and another bunny not bad notified of money all it could be 1 type of Battenberg cake and another type of Battenberg cake in which case I get the iterated Battenberg cake and this is quite prefer advanced piece of abstract months that Eugenie see if you're a 2nd year undergraduate masters student so I say why not show it's where 1 anyway because it's kind of cool and here is a very abstract representation of Battenberg cake so all I have to tell you actually was 2 pieces of information what goes in position 1 and position 2 so it could be yellow and pink it could be topic more chocolate it could be body 1 body 2 or to be take 1 and cake to but what is take 1 take 1 is already 1 of these and take 2 was already 1 of those so when I put those over there where it says 1 and put take 2 over there in the 2nd position and this gives me an extremely abstract representation of about and iterated Battenberg cake well i've basically removed all the information apart from what I'm putting in and so this is now a tree it's what mathematicians study and they call them trees and trees come from sticking things together because in novel life they start at the bottom and they grow up splits out but in that's life you can read them upside down because the abstract so you can read the list of ingredients of the topic here sticking together in certain ways and I love sticking ingredients together you can do things like have this distri can compare with battery so this tree says this put the 2 things on the left together 1st and best tree says put the 2 things on the right to get the 1st so for example we could do for outflow which is the 8 mostly and then we could add to on afterwards and that makes 10 and you because we stuff things into boring straight lines we have to write it with parentheses like that and the other way around would say do fall add 2 which is 6 and then add the other 4 afterwards which also makes 10 because we know the addition is associative and is usually a some kind of stupid rule what we tell students in that you have to manipulate these parentheses around it's much prettier with the trees I like leaving things in their natural geometrical representations rather than stuffing them straight lines it on there are situations where sensitivity doesn't hold for example if you commit and yolks with sugar and then you put milk in and you heat it up a bit you get custard which is 1 of my favorite things if you do it the other way round you next sugar and milk together and then you put the egg yolks in off to if you don't like that you do not get custody and in in that but mathematics you have things that are much more interesting than just equations and the reason is your studying things that a much more interesting than numbers so things can be not necessary just equal and not equal but they might be slightly different very different extremely different for example if you make cake usually the recipe says screen as shown in the bottom add the eggs and then fold in the flour you make a it but actually it turns out if you can just kind of if you because you have an electric which you can just kind of talk everything in a ball mill are are basically the same and you can try all possible ways of doing it as well and all possible different trees look like this for for ingredients so this says that this is how I'm going to move 1 little branch from left to right each of these arrows you could see that it's moving wonderful brought from left to right and each 1 represents a different way of combining 4 ingredients cake so this 1 says take a b of parts and the eggs mix them together 1st which looks really discussed and then add the sugar and then add the flour B and actually if you beat a hard enough at the end it doesn't really make a difference I did this with my art students we broaden reason we tried it and the thing is that in the end it looks the same the outcome is the same but the process is very different so it just depends what you care about the process or just the outcome and math is more about the process in the end than just the outcome which makes it more it makes it more than I the most subtle so if you are making a cake do you can for example at parity your paper looks like vomit in the middle because that's what happens if you just mix X with butter this 1 here you have to make the bottom the sugar on 1 side and you have to make the eggs in the file on the other side which not only looks disgusting but it means you need 2 balls because then you have to put you have to make those things epidemics them together often if you have 2 balls that means more balls to wash up and if you're lazy like me that makes a difference if you're going to eat the cake at the end there may be doesn't make a difference so it's up to you to decide what counts as the same or not and I think giving people the opportunity to make more choices of their own when they're doing mass gives them all ownership over and keep them engaged so finally I want to say well what happens if you want topic case which I usually do then you need 1 more ingredient and that means you need to study all the trees with 5 leaves and if you do that on the you can separate them out into lots of different pentagons the fit together in certain ways and toward the end of my but there's a net resource and that's already of there's a net of what you can cut out to make it but somebody showed up to 1 of my talks and had 3 D printed me a version of the shape you get photopic cake and they they but this little hook on it so I could hang on my tree as a Christmas decoration in this festive color black but the the this is there something that you usually only see when you're doing a PhD probably in category theory whereas it's really just all the different ways of sticking topic a together so I say why don't we make more of the things bring more of the fun things in earlier and not say that you have to pass millions of exams before getting to see this here look you can see it's already without passing any exams and you can appreciate it in many different ways at many different levels so that's why we propose as a way of communicating that to people thank you hi how OK that already for questions the the so the year how many Battenberg cake so there in the fall I fall I started at least 1 more how many this is a so you can sort of D 1 of 3 and the ones in the trees have any other I don't know of so do you do mathematics that you can actually studied that is to say when you eat it then it makes you reflect magic I like to think that anyone eaten my iterated Battenberg cake reflects upon mathematics while they're doing it but the real
actually came about was because my 2nd year students who were really mad students they could never remember the difference between the 2 of groups of order for the Klein four-group in this like a group of order 4 so I made these 2 cakes for them and then they remembered forever yeah I there have to look at it don't didn't use you have to look at it so something that we that through the use of means the all the reason of the things that you can see the reason of the things that you can hear that you can touch but use them of the things that you can please I think there is so I did another thing with my students well actually there was 1 thing where some assume I'm brought putting in an American pudding is like it's kind of like moves and she brought this putting in and then we ate the putting in at the bottom she had placed a proof because the proof was in the putting on but I know I I I have demonstrated their structural things that their that their structure things we take the recipes and you kind of put them together and then in their the place where they meet they kind of blend so the 1 I have tried to do to exhibit this is a as a cross between a talker Brownian cookies where you put all your chart your brownie that's in it in any block chocolate chip dough took to cookie dough on top so you have brownie and cookie and then in between a kind of blends into each other so you can see how you can take 2 different types of ingredient there's another 1 I do which is kind of profound this piece of number theory where I still have blueberry jam into yogurt which is like a joining in a rational to the integers to because as the difference between that because when you do that if you know understand what that means if you start a read to into the integers than some of the route to turn back into integers is when you multiply them together the root 2 times the route to become the 2 again whereas if you start brussel sprouts into your that the brussels sprouts do not blend with the yogurt all and that's that's like adjoining an X to the integers and making polynomials because if you add an extra bit integers then you multiply the exit together they never become integers again they just become more and more things and my students always got mixed up between the gas in integers and integral polynomials as I showed them this demonstration showed that when you stubbly real that damage your that your that returns purple which means that some of the blue breeze really go back into the and that exhibited to
them so maybe that is the way you can taste the mathematics FIL the as we but Diablo Hollis and fortifying uh those with 0 Ituri connects to mass so as of nest present her right or working like I really need to fight a those lots of examples and I was really as a a curious on the surprising to find all those but you're examples in this news of I was for a b d have special holes and finding that but I'm glad that you liked it think that's what I heard anyway and the so the thing is that because I truly believe that maths connect but I truly believe it connects everything and I
truly believe that there is some maps in everything and that my way of thinking about the world is very mathematical I can look at anything and find something mathematical say about that because I actually believe that I can anything ularies that happens to be in the world I can turn into some story that relates to a mathematical way of thinking and it's because I've noticed that that's the trouble with abstract masses pick this abstract and so the same thing that makes it powerful is the thing that makes it difficult to explain and so if I connected it with something hilarious the hand in my life that people can relate to or taste or touch would have some kind of emotional response to like maybe some people so offended by the idea of making take a different way but the even being offended despite having some kind of emotional response is the key because then it connects to somewhere inside you instead of being just some dry thing with someone's status that's writing at a bold and droning on that as long as it connects to some emotional part of students then it will wake them up to some possibility and and also that remember at more so basically everything everything I encounter that some sometimes some students and that 1 time I showed up to class and there was a plate of random Oreo cookies sitting on the front desk of my students challenged me they said because life cycle they brought me or cookies maple I have brought them moral cookies none of us abroad because we never found out what the oracle is doing and then some his challenge me instead use it explains a map and so I started it and all its conjugation and have him which must know but we were doing conjugation groups which is where you you take an element and you put 1 element on 1 side then the inverse of it on the other side and that's warrior cookies or the previous and have a cookie on 1 side and the 1 that the applicant unified upside down
and when students do conjugation they always forget to invert the element on the other side and so this is 1 article accusing and they never forgot what conjugation what they sometimes called Oreo cookies in exams instead of conjugation but that's OK the it 1 more question here of just sort of interest and what was set on the Pentagon 1 of ornaments but as ornaments it's called socio it's something that comes up in topology in high-dimensional category theory so that and they the more every time you add another ingredient you get another dimension so if you wanted to have a cake with 6 ingredients you need a four-dimensional object which we don't have a four-dimensional printers so much yet but it has 6 pentagons and 3 squares so he did the next 1 up it would have 7 of these and then the next 1 out of 2 that would have 8 of those so you can do you can do it dimension shift so that the Pentagon is called the McClane Pentagon after you did it the associate Heeger were done this that they would dreamt up in invented discovered whatever by extension studying sin and topological things to do with how badly things fail to be associative and on how far they are from BIC boisei to students if you're if you there we study how how badly things fail to be true and then we study all the different ways of failing and I say to you that they all fail in many different ways as possible and then classify all the different ways of bailing and see whether they were really different woman during which we've 900 here thank