Paul Zeitz, who's one of our visiting mathematicians, is also going to be introducing tonight's speaker. And just to tell you a little bit about Paul, he has been a longtime supporter and friend of this museum since before it even opened. I first got wind of Paul when we sent one of our traveling exhibits to a museum in Berkeley, California, and someone told me there was this guy coming in and running programs with our exhibit and we sent a good guy except we hadn't sent anyone. And it was Paul and he just loved our exhibit so much that and he loved the math of the exhibit and talking to the public about it that he started interacting with the public around the exhibit and became really a good friend to the museum. He also happens to be one of the founders of the proof school, which is a math school out in the Berkeley area, one of the founders of the Bay Area Math Circle. He's an author and he's also a former international math Olympiad contestant as well as now a coach as well as the person who's organizing the IMO that will be here in this country maybe in Washington DC in two years. Okay, got that right. And so Paul's a great guy and we're really delighted to have both Peter and Paul with us for the year and with that I hope you'll join me in welcoming, yeah I know we don't have Mary. If anybody knows a female mathematician named Mary who'd like to be here for the year, we'd love to have her because that would just be great. But in the meantime Paul is going to introduce tonight's speaker who is unfortunately not named Mary, but she's nevertheless going to give a great talk. So let me turn it over to Paul. Thank you very much. So I'm here to introduce Cornelia Van Cot and she happens to be my colleague. In fact she's sort of my boss. She's the department chair at the University of San Francisco where we both work and I've known her for more than 10 years. We hired her 11 years ago and so let me tell you a little bit about her. She is from Indiana. She grew up in Indiana. She went to college in Illinois but that was too far from Indiana so she went back to Indiana for grad school where she studied topology and became a not theorist and she does very exciting research. Recently she's been discovering some connections between not theory and number theory. We hired her at the University of San Francisco and now she's our department head. Now she also plays the church organ for fun and like anyone from Indiana she is a very, very serious basketball fan. So these are just sort of facts and they don't really tell you anything about what Cornelia is like and so something that's important about math that many people don't understand but maybe you do if you come to the museum is that mathematics is a social endeavor and in fact Ravi Vakil who was a speaker here two years ago he told me about 12 years ago that he reminded me that mathematics is the most social of the sciences and the reason for that is because math is really difficult and so people need to work together in order to penetrate mathematical mysteries. So 11 years ago when we were interviewing Cornelia for a job at the University of San Francisco one thing she told us was that she had organized a conference for for young topologists and she boasted she said I know all the young topologists and she meant that literally a year later after we had hired her she was on a hiring committee with us and we were at a convention in in Washington DC and I found that it was impossible to walk more than 10 feet with her because somebody would run up to her and hug her and so every 15 seconds hug it was like being with a celebrity and and the reason I'm telling you this is is to remind you that what makes mathematics important is not the math itself but the ability to share it and to communicate it with others and and that's why I'm really grateful that Cornelia is my colleague and that's why I'm thrilled that you're going to experience some of the mathematics that she wants to share with you tonight. So thank you. All right thank you very much for the kind introduction and thank you all for coming on Wednesday night to come here more about math and our topic today is Pi and if you read the title Pi sometimes equals four you should feel that that's a little bit of mathematical blasphemy right because if you you may have forgotten a lot of things from high school math but you one thing that usually remains is people remember that Pi is 3.14 or something like this and your feelings of shock are reminiscent of this scene from The Simpsons which I'd love for us all to recall or or hear about for the first time if you haven't seen it. There's a scene where Professor Frink is trying to get this room full of scientists to quiet down and they're all chatting amongst themselves no one's paying attention to him and he's saying order order scientists I need you to give me some order no one listens so finally he raises his voice and shouts out does anyone know what he shouts out. Pi is exactly three and immediately the crowd gasps and falls silent so you have Professor Frink saying Pi is exactly three you have me saying Pi is sometimes equal to four and at the end of this talk you'll be able to evaluate any such statement. All right but let's first push away these blasphemous statements no more of that let's go back to simple beginnings how was Pi originally defined and what we all know and love from high school math. We know that Pi is 3.14 and then it just goes on and on forever. Pi is one of those mysterious irrational numbers the decimal digits never end and they never become periodic so you can see that the digits are seeming complete jumble. All right and even if you've left high school and you aren't really connected with math anymore Pi still comes up in your daily life maybe once a year because there's this new trend to celebrate Pi day on March 14th. All right so maybe even once a year you get reminded of this value but one of the things that Americans can do is they start to celebrate things or get into trends and they forget the reason why they were interested in these things from the first place for example Labor Day do we still remember or are we just happy for a three-day weekend yeah but we don't want that to happen to Pi day may it never be right so we need to remember why is this number unique what makes it special why aren't we celebrating say 2.14159 besides the fact that maybe that would mean we have to celebrate on Valentine's Day and that's already taken. Okay but there is something really beautiful and mysterious about this number let's recall what it is it all comes from circles so you draw a circle and you measure two things about the circle measure its circumference and its diameter take those two numbers and then wonderful thing is that it doesn't matter what circle you choose small or large this ratio will be the same it'll be that very mysterious irrational number Pi that's where the first place where you notice you know this Pi number it's unique it shows up as this ratio no matter what circle I draw so you might have maybe a handful of these digits memorized yourself but but you know these digits they just go on and on with no rhyme or reason and it's a fair question for you to ask how do mathematicians know that like these digits are all correct and if I just left the room and said you guys need to give me the next digit what would you do to find that next digit so I'm going to tell you how mathematicians go about finding the digits of Pi just to give you a basic idea so it's not through measuring circles extra carefully anymore that was how it was done originally through antiquity and you measure very carefully measure across with with very clever things but now we don't do this we get at Pi through a different direction we have formulas clever formulas for Pi and they help us get an estimate let me give you one such clever formula to start so we want to first take all the odd integers 1 3 5 7 9 and the dot dot dot means just keep them on coming it's an infinite list of odd integers and now take the reciprocals of all those odd integers that means put them all in the denominator of a fraction okay next step is we're going to turn this into a big long addition problem and make plus minus plus minus every other number so there we go so this is an infinite addition problem that's it that's a very formidable thing but we'll just skip right over that for now and we'll take that whole sum and multiply it by 4 and what do you get when you do that you get Pi so this is a clever formula for Pi it has some concerning things that this is an infinite sum but it's actually very famous it's called Leibniz's formula and we can use this to get an estimate on Pi so you can't you can't in human time add up all these numbers in this parentheses there's infinitely many of them but maybe you could just add up the first 100 numbers multiply by 4 and you get an estimate for Pi and if you want to improve on that maybe add up the first 200 numbers multiply by 4 get a better estimate for Pi so in this way we get closer and closer and figure out the digits of Pi but unfortunately this isn't quite as useful as you might think so to get 10 decimal digits of Pi you must add up the first five billion terms so this formula though it looks very friendly in some sense it's not too complicated it doesn't give you a lot of bang for your buck you have to add up five billion terms to get the first 10 decimal digits there's there's there's too much more to go sorry so let me show you we don't use this today we use something else it's the Chudnovsky formula and it was discovered by these two brothers they're the Chudnovsky brothers they they live here in New York City and they immigrated from the Ukraine and here's their formula and it's gonna look really scary compared to the previous one so here it is looks really messy but if you if you can just parse it just a tad it's really just a big fraction right and in the numerator we have this number and in the denominator we have an infinite sum just like on the previous equation we're supposed to add up an infinite number of terms if you're not familiar with this notation that's what it represents you're supposed to add up an infinite number of terms and it works like dynamite compared to Leibniz's formula after summing just two terms you already have more than 10 digits of Pi okay the Leibniz's formula you had to do billions of terms here just two terms and with each further term you increase the accuracy by 14 digits on average so you're just galloping through getting more and more digits of accuracy with each further term that you add in in this infinite sum alright so this is a big improvement and and before we move on I just want to mention that these two brothers have a really interesting life story and it was written up in the New Yorker called the mountains of Pi because they they discovered that formula and then they decided to build a super computer in their Manhattan apartment to compute these digits of Pi and yeah they just made it through mail order parts they they said that they needed to give acknowledgments to Home Depot for all their all their help and making their supercomputer the story is very interesting so on your subway ride home you should look it up and and enjoy reading their interesting lives but I I also want to highlight that this is the formula that's used today it was discovered in the 1980s late 1980s but just this year this is a picture of Emma O'ahu she's a Google employee and she used the Chudnovsky formula to compute the first 31 trillion digits of Pi and made the world record of course and she didn't just use this formula she also used Google Cloud which provided a lot of data storage so the two things together made for the ability to to get that far into the decimal digits of Pi all right now I want to go back and talk about something that I kind of glossed over before and that is something about circles so when you read those instructions there at the top of the of the screen it says you're supposed to draw a circle of radius 3 centered at the origin and most of you have great muscle memory and memory from how things should work and you you could draw this shape no problem but what I want to highlight that maybe you don't think about because you've done it so much is that the definition of circle depends on how you describe distance so for example this point up here he's part of the circle why because the distance from the origin up to this point is 3 so he gets to be part of the circle next this point is part of the circle because the distance between the origin and that point is also 3 and so really it's a point-by-point discovery process to say who gets to be in the circle and you're measuring distance from the origin to the point by a straight line and eventually you realize oh I see they're all these points and they're all part of the circle but I just want to highlight that the definition of a circle depends on finding all the points whose distance from the origin is a particular value now I'd like to remind you of a good principle for being a mathematician is that you should have a rebellious edge if someone tells you you should do something this way this is the rule in math you should always say what happens if I break the rule it should just be part of how you think it sounds bad but when you do that one of two things will happen either you'll realize oh whoops there was a good reason for that rule and you'll go back and you'll be a stronger mathematician because of it you'll understand the reasons why you're doing things or another possibility is you'll discover something new all right so we're gonna try the same sort of thing here I told you something just now and probably everyone swallowed it hook line and sinker and I'm gonna help you develop that rebellious edge the thing I said is that the distance between two points is measured by a straight line between them and you probably all feel like that's a great idea right and you probably even remember the formula for the distance between two points the distance formula this is a nice choice it's called the Euclidean distance and it's a good choice if you're a crow because it's like as the crow flies or if you own a helicopter but there are lots of other ways to measure distance let's choose one and see what happens so away away it goes let's find a different distance function so the most common alternative is something called taxi cab distance so you imagine that there's a car at point P and he needs to get up to point Q and it can't just drive rough shot over people's backyards and pools no he needs to stay on the road and by staying on the road in this context I mean the car can only drive on horizontal and vertical lines so to get from P to Q it travels over and up okay so it's a new way of getting from P to Q not by a straight line but by traveling over and up taxi cab distance we come up with the formula for this how far did the the taxi cab drive horizontally well it's just the difference in X values and how far the taxi cab drive vertically difference in Y values add those two together and you get the total distance traveled this is also called the Manhattan metric because it models driving around in Manhattan to a point all right so let's adopt this like probably none of us like the square root and the old distance function right it always caused numbers to come out strange this has no square root we're liking this a lot so let's choose this as our distance function and see what happens the first thing that changes is the shape of circles so let's consider what would it look like to draw a circle of radius 3 about the origin with taxi cab distance all right so you got to imagine putting that little taxi cab driver right at the origin he can only drive a distance 3 where are the places he can end up well he could drive straight the distance 3 so there's one point on the circle we could go vertically and there's another point on the circle and he could travel over one up to there's another point or over two up one there's another point on the circle and so on and you don't have to just go into your values you can do anything so I could go over a half up two and a half and there's another point on the circle and you can see what it's tracing out here we're going to end up with this entire diagonal line of points or points on the circle and similarly if I let the driver drive to the left or downward we trace out this shape here okay so that's a circle in taxi cab distance all right so so I'm considering can I figure out the the circumference and the diameter of these and and compute that so let's first reduce down to the unit taxicab distance unit taxicab circle so it's radius is 1 and consider what's the circumference divided by diameter here now there's a lot of wrong things that might seem really right right now there's some very tempting things you could say about how long these these diagonal line segments are for example I expect for many of us it's tempting to say that the length of this diagonal line is the square root of 2 why is that tempting because what do you see right here a right triangle and you can use the Pythagorean theorem so the length here is a square root of 2 but that's if you're going as the crow flies right but we are not doing that anymore we threw that away and we have a new distance function so you have to imagine the taxi cab driver trying to travel along this line all right so that should make things seem a little shaky right now so here's what I'm going to do we're going to go back to Euclidean world and figure out how did we do circumference there and then we'll come right back here and and apply the same method here so back in the Euclidean world when we have these circles that feels more normal right how did we find circumference here well there was something tricky about this situation because this this shape is nice and curvy and we are trying to measure distance with a straight line so there's some dissonance how do you take your straight ruler and measure the distance around a curvy shape so this is a problem that mathematicians have been studying throughout hundreds of years and here's how they approach it you start out with a circle and you can inscribe in it polygons and you notice as you add more and more polygons the perimeter of these polygons is getting very close to the circumference of the circle and the perimeter of the polygons is something that you could all compute because they have nice straight line edges okay so through computing the perimeter of these polygons we can get an approximation for the circumference of the circle okay so the short the short tagline here is that we approximated this curvy shape with straight lines all right take that thought and let's apply it to our taxicab world we no longer have a curvy shape for circles but we have a different problem now the circles have diagonal lines and we're supposed to be measuring distance with horizontal and vertical lines so we still have a rub something that doesn't quite work out but we'll apply the same principle we'll try to approximate this diagonal line with horizontal and vertical lines let's zoom in on that first quadrant just zoom in on that there's our diagonal line we put our taxicab driver there we'd like him to travel in a path that approximates the diagonal line with his usual horizontal and vertical lines here's a very rough approximation the taxicab driver can drive up and over we can do better than this let's let's make it a little better there's up and over with two stair steps and we can keep improving allowing more and more stair steps there's four we can keep doing better we can keep doing better so do you see how this is the same kind of thing as the polygons inside the Euclidean circle we're putting more and more stair steps along this diagonal line all right and the principle is that the length of the stair steps the distance that the taxicab driver has to drive to get to the top is going to be approximating the taxicab length of the diagonal line but in this case something very beautiful happens and that is you might say well the distance that the taxicab driver drives depends on the number of steps like what if there were a thousand steps or what if there were two thousand steps that might seem true but actually it doesn't matter if there's four steps or if there's a thousand steps if the taxicab driver drives along those steps he will always travel the same distance could anyone contribute what that distance will be really loud too thank you yes I'll show you why that's true if you just have four steps and it will work for any number so here's four steps how can I see that this driver will drive a distance of two if he drives on those stairs you can see that all the up parts of the stairs if I slide them over there a total distance up of one his vertical distance traveled is one and all the horizontal parts where he traveled they can all add up to a total horizontal distance of one so he travels a total vertical distance of one horizontal distance of one a total of two and if he had had a hundred stair steps the picture would it look exactly the same it just would have maybe taken three hours to create right all right so that's great we just decided that the taxicab distance of this line is two and we can copy it around because these are all the same just symmetric and now we're to the point of the talk where everyone can contribute everyone even if you feel like you weren't sure about it I think that now you can tell me what's the circumference of a taxicab unit circle if he traveled all the way around how far would he travel say it loud eight thank you and then if the taxi cab driver had to drive across the circle diameter how far would he travel to all right let's plug these numbers in eight over two equals four so that's why we say that the value of pi when we choose taxicab distance is equal to four okay so you see that domino effect of changes we decided to make that change in our distance function that changed how our circles looked and now it's changing the value of pi to four all right so the emotions you feel right now may be mixed because pi had been this mysterious number right there's nothing mysterious about four the old pi had infinitely many digits four is an integer right so you feel it's not quite as mysterious as before not quite as interesting maybe but in fact you should consider the flip side which is huh you know what I changed my distance function and circles turned into these diamond shapes what other shapes are possible like choose your favorite shape you'd like to show up as a circle and and we'll find out if it's possible and then I mean pi just ended up being four two minutes ago what other numbers could show up pick your favorite number if we just change the distance function again and the beautiful thing is that mathematicians can answer this whole question fully and by the end of this talk you will know this whole answer fully but before I reveal the answer I think it's important that we realize that not everything works out quite quite like a charm some things are difficult so I'm gonna make us do something difficult first maybe not difficult but maybe it doesn't work out how you thought so here's something fun we're gonna measure distance differently we're gonna measure using discrete metric so forget about Euclidean distance forget about taxicab we'll do discrete metric and I have nicknamed this also the teleportation metric so here's the story to help you get the formula let's say I would prefer to be in London right now I can just snap my finger and teleport and be in London or if I'd like to be sitting on the front row over here I can snap my finger and be there or if I'd like to remain here and keep on giving this talk I don't have to snap my fingers at all so what's the analogy here with distance you have two points how far apart are they well the distance is one so you can just snap fingers and teleport from one point to another even if they're really far away or really close all of their distances are one because one snap will get you there and if you have a point to itself how far is a point from itself zero because it doesn't need to snap the fingers I hope that makes sense and here's a formula that just says what I just said the distance between two points is either one or zero there's only choices it's one if the two points are different it's zero if the points are the same all right so I'd like for each of you to think for yourself what do circles look like here so consider what would a circle centered at the origin of radius one look like and what would a circle centered at the origin of radius three look like and there's the formula and I'd love it if you could get your graph paper out and sketch what you think it looks like in both cases show your neighbor to so that you can have a little bit of conversation about it we'll give you about two minutes to try it out and if you're confused or have a question raise your hand and I'll come help you all right let's pool all of our discussions together and see what we have so in this picture you need to find all the points whose distance from here is one so what do we need to shade in so everybody is a distance one right this point is a distance one this point this point this point is there anyone who's not a distance one from the origin yeah say again the origin itself he's a distance zero so we just need to shade in everything except for the origin how's it work over here I need to shade in all the points who are a distance three from the origin so so is this point how far is he from the origin is he a distance three no he's a distance one okay so are there any points I need to shade in the point itself how far is the point from himself zero right he's not three so let's see it's a doughnut well I would actually say that there are no points you can't name for me any point who when I say how far is he from this you say three because if you tell me oh this point is part of the circle I say oh but his distance is one not three yeah no actually there's no points like even if you took something way out here how far is he from here one still one yeah but that's it's kind of shocking right it's like not the answer you thought you were supposed to have so on the left we color in everything except the origin and this one is awkwardly empty there's nothing to color it and it's like a trick question right like you're looking for something to color in but there's nothing okay so just like before let's let's carry on like soldiers what's the circumference divided by diameter in either of these cases so if you're a normal human being you feel like this is not something you can think about it seems undefined so and you're right this wasn't this didn't work out nearly as well as the taxicab one did right the taxicab we worked and we succeeded this one we worked and we are gonna give up because we realize this isn't the type of distance function we're interested in okay so let's switch it up now let's try the post office distance or post office metric I'm gonna tell you the analogy here and then you'll consider what circles look like let's say I need to send you a birthday card if I'm sending you a birthday card even if I live right next door to you what's the path of that birthday card it doesn't go right next door it goes all the way to the post office and then comes back to you so we use that analogy it doesn't work perfectly but use that analogy to understand this formula for distance in post office metric goes like this to get from P to Q imagine it like a letter can't go directly from P to Q so where's it gonna go first where's the post office do you anticipate the post office is at the origin and so the letter goes directly to the post office that's the one place where it doesn't quite follow the analogy because it's like the post office has a helicopter or something goes directly to the post office and then directly out to Q alright so keep that picture in mind to get from P to Q you have to come into the center and then go out okay that's how you get from one point to another we can give a formula for this there's our formula it looks a little complicated but this is just saying find the distance from P to the origin plus the distance from Q to the origin and if you're sending a letter to yourself you don't need to use the post office system so the distance from P to Q if P equals Q it's just zero you don't need to get involved alright so is everyone feel they can know how to get from one point to the other and this distance function hope so so now I'm gonna give you another exercise here's a point P equals 4 3 I'd like you to find the circles of radius 3 5 6 and 10 centered at this point so let's kind of talk through one of them together so you can imagine you have a letter and you put just enough postage for your letter to travel a distance of 3 who can you send a letter to nobody someone said nobody and you're correct this is that another trick question kind of the one you were talking about because you can't even get to the post office in distance 3 right so you can't send a letter to anybody so this circle in this case will be empty no points at all okay now you've gotten a little push a little head start and now try to figure out 5 6 and 10 yourself in the next few minutes on your own or with your neighbor even better okay let's pull all of our results together so so far we already talked about what happens when radius is 3 we can't even get to the post office so the circle is nothing there is nothing to color in it's empty set alright but how do we do how does things work when the radius is 5 who can we send letters to anyone the post office yeah you can reach the post office but you've already used up all your postage there so you couldn't talk to the post office so this dot here is a circle centered at P of radius 5 good job okay what about radius 6 what will that circle look like any one shout it out say again yeah a Euclidean circle centered at at the origin of radius 1 that's what a unit circle that's what you said yeah so so you can get to the post office use you've already used up distance 5 so you still have one distance one left so you can travel around the distance of one so that is a circle of radius 6 centered at P alright now we move it move up to 10 how does this one work out is there anything different about this one than the last one or is it exactly the same yes someone said P is not on the circle very good so if I if I just went like before I use up distance 5 just to get to the origin I still have distance 5 left so I would get this circle Euclidean circle of radius 5 centered at the origin but P itself how far is P from itself 0 it's not a distance 10 from itself so he doesn't belong in the circle so you cut it out so so this is a circle of radius 10 centered at P okay now we've reached the extra credit part of the lecture where we think about what's the circumference of this circle and the reason I make it extra cut it is it might be it might be a little overwhelming it's not quite as as routine it looks friendly enough right this looks kind of like a Euclidean circle so it seems like the circumference should be something normal but it's not it would be normal and and friendly if we were measuring distance using Euclidean distance but we have to imagine a post office worker visiting all these points along the circle and how does the post office worker travel he needs to go in and out in and out in and out so what's his what is the length of his path going to be it's going to be infinite yeah so the circumference doesn't work out in this case either so I've just handed you two distance functions neither of which worked out nearly as well as the first two that I handed you but that's good because now we see that our expectations were a little high we thought that there'd be this nice flow chart you start with a distance function it gives you circles and then we get a value for pi and now we've got to amend that and that's good thing that you can't just start off with any distance function it's got to be a good one okay not everybody works so I'm going to tell you exactly what the rules are to be a good distance function mathematicians I figured this out and some of the rules you'll say of course I would never think of breaking that rule but we just want to make sure we list everything so no one tries to get anything past us so the first rule is that your your distance function must be positive definite what that means is you cannot say the distance between two points is a negative number that's outside of out of bounds okay that's an example of something that you probably wouldn't ever do so we have to say this value the distance between two points P and Q must be non-negative it could be zero the only time it can be zero is if the two points are the same and other than that it has to be positive the next requirement of being a good distance function is that it must be symmetric that means that if you measured the distance from P to Q whatever you decide that should be right there's freedom there but the distance back from Q back to P needs to be the same number you can't be so creative that the distance to one in one direction is different in the distance back okay that's second next one is called the triangle inequality it's a really famous requirement and I often call it the ice cream rule for the following reason so we have two points P and Q and just to help you bring it down to earth let's imagine P is here we are all the talk and then after the talk you're gonna go home and and you could after the talk just go directly home that's fine but there's always an option that you could stop off for ice cream on the way home so there's two possible paths to your home going directly there or stopping off for ice cream so if you stop off for ice cream what will that likely do to the length of your trip home it'll likely make it bigger so that's what triangle inequality says is that if you stop off for ice cream on the way home it'll make your make your path grow in length it's possible it could be equal in length what are the what would make that happen you could live at the ice cream shop then it would be equal or the ice cream shop should could be directly on your way home right so here's the here's the triangle inequality and in mathematical language that the distance between P and Q MoMath and home is less than or equal the distance between going from MoMath to ice cream and then ice cream to home all right next for next requirement to be a good distance function is it must be translation and variant so that means if you take two points and you translate them both so they both move with the same motion to two new points P prime and Q prime the distance before and after translation must be the same the distance between P and Q is must be the same as P prime and Q prime now we're ready for the final requirement on what it means to be a good distance function and it's the most subtle not quite as easy it says if you take two points P and Q and you do a scalar multiple of both of them so multiply both of them by some scalar we'll call it R you end up with RP and RQ so it just dilates them out or shrinks them in and this requirement says that the distance between RP and RQ is just equal to R times the distance between P and Q that one's a little harder but it's necessary all right so here's the list here's the list of all those five requirements and and remember the post office distance was not a good distance function so that means the post office distance failed one of these requirements can anyone take a moment and maybe take a guess of which one did the post office distance fail you it definitely fails for if you take two points and translate them both up did they just stay the same distance apart no it went up they're now even further from the post office so it takes even further to get from one point to the other so it's not translation and variant fails for it passes five the discrete fun the the just the discrete distance function fails one of these as well the teleportation one it fails five yes it actually passes all one two three and four you can think through them why it passes all the first ones and fails five okay so we're gonna forget about the bad ones no more like erase the post office distance from your head for now and we just want to focus on good ones that make this beautiful geometry happen and good distance functions have a mathematical term they're known as norms norms on the plane so we'd like to find a few more norms on the plane so far we just have to the Euclidean distance function and taxicab distance we'd like a few more if I asked you to come up with a few more this is how this is how you would feel you'd feel like this guy trying blindfolded and groping about to try to find another norm it feels impossible why does it feel impossible because you're trying to come up with a function that satisfies five different requirements all at once it it it doesn't doesn't seem like it's gonna happen so I'm gonna get you started and introduce you to your first family of norms it's a very famous family and the two norms we've already met they are two members of this infinite family so here's Euclidean distance its formula here and here's taxicab distance and here's its formula here and here's my plan I'm gonna rewrite this distance equation in a way such that it mirrors this one up here not gonna change it just rewrite it in a different way so I'm gonna erase and rewrite it this way still the taxicab distance formula but do you see now how it looks really similar to this one and you can see now how you could generalize to think of a next distance function that would like fit in this lineup hopefully you can and these are called the LP norms so instead of ones and twos we put any number P where P is a any number greater than or equal to one so P can be two three four five it can even be non integer values like six point eight or pie it can be any number you'd like and so now we have infinitely many good distance functions or norms and now we want to see what the unit circles look like for LP norms so here's the unit circle when P equals one it's the taxicab unit circle those those diamonds we make we constructed before and now I'm gonna let P increase because it's supposed to start at one and be any number bigger so we're gonna let P increase and see what happens to the unit circle and you'll see that they'll bow out right now it's really nice and tight they'll bow out so here's P equals one point two P equals one point five here's P equals two that's Euclidean beautiful Euclidean circle and then P equals three P equals five P equals ten and you can see that they're limiting if you take the limit as P goes to infinity you get the unit square so that's what these are a whole bunch of new circles with different norms now we can figure out what the pie values are associated to them so here's the theorem for that this is a theorem of Adler and Tanton many people proved this independently but this is just written up very nicely and kind of recently so I'm quoting them they said let pi sub p denote the pi value associated to the LP norm then pi sub p that pi value is in this interval between our usual value of pi and four let's just look at the picture the pi values of the LP norms fill in this interval between our usual value of pi sweeping out all of these values all the way up to four rational irrational everything is swept out by the LP norms okay so we just now added infinitely many values that pi can be sometimes pi equals four but sometimes pi equals three point five right or sometimes pi equals three point eight eight eight all of those things and you could ask like pick your favorite number in here say three point five say how many LP norms have that value as their pi value the answer is exactly two and here's the theorem for that it's a theorem of Keller and Vakil in 2009 they said that two pi values are equal for two LP norms if and only if one over p plus one over q equals one so for example the L three norm and the L three halves norm they both have the same pi value that's because one over three plus one over three halves equals one all right so far what we've done is great but to be honest we've only figured out how things work for a very small subset of all the norms on the plane we have a whole lot of more norms to tackle and if we had to keep going in the same way like coming up with formulas then drawing the circles then computing the value of pi then we would all kind of lose heart because it's not so interesting but this is an exciting and amazing part of the talk that we don't have to continue this way we can push it all away and solve this entire problem geometrically and I'll explain exactly what I mean so so far this is the the struggling part of our existence so far we've been coming up with a formula we work really hard then we draw the unit circle and compute the value of pi but we're going to turn the tables around and say you come up with what you want the unit circle to look like just draw something as long as it satisfies three geometric requirements then that's fine that will uniquely determine a norm they're in one-to-one correspondence okay so instead of banging your head against a wall finding a formula that satisfies five requirements you could just draw a picture of how you want your unit circles to be satisfying three visual guidelines and then you're in business okay so no more of this we're gonna work this way we want to find good unit circles they'll uniquely define a norm and we'll have a pi value as well okay so what are those three requirements on unit circles that make them good the first requirement is it must enclose a finite area and include a neighborhood of the origin so you can't draw a unit circle it goes off for forever like the discrete metric that's not that's not okay and the second requirement is if you connect any two points in the shape with a straight line the line must stay inside the shape there's a math word for this and it's called convexity that's what this is exactly what it means and the third requirement is that the shape should be symmetric under 180 degree rotation so if I take the shape and rotated 180 degrees it should look as if you hadn't done a thing it rotates right back into itself with 180 degree turn okay so if first if out there there are a few of you that said I would really love it if we could find a distance function where the unit circle looks like hearts wouldn't that be cool your heart is broken now because you now realize it will never happen it which of these things does it fail which of these requirements does it fail it fails it fails to and three so it fails to because here's two points on the heart and if you connect them with the line the line does not stay inside the heart so it fails to and it also fails three if I took this and did a 180 degree turn it would be an upside down heart it would no longer look like a heart okay there's another shape that we have to lay aside and that's a triangle what does it fail it fails three so it actually is fine it is a convex shape these are these are totally fine but it feels three because if you rotate it around it's now upside down it doesn't look the same all right but never fear there are lots of shapes that do fit these requirements here's a bunch of them I just put a bunch but you could come up with many many many many more and I have two things left to describe to you given these given a shape like this how does it uniquely determine a norm and what sorts of pie values show up all right so first we'll describe how do you determine a norm from this if you can keep your bearings about you I know you've been through a lot so far in this talk but but but just because you decide what a unit circle looks like how does that tell you the distance between points that's what a norm is telling me with the distance so we're going to do it with this example let's say I want my unit circle to look like this shape how would that just determine the distance between two points so there's my beautiful unit circle doesn't it look great and and here's two points P and Q how could I determine how far apart they are so there's a process and that it's just a three-step process and I'll just describe it all in one side here for you the first step is to take the two points and find P minus Q that will be another point on the plane and in this case there's P minus Q and the next step you draw a ray from the origin through P minus Q there it is and that ray will cut through the unit circle in a single point and we'll call that point you for unit circle now let's look do you see you and P minus Q they are multiples of each other if I just stretch you out it would equal P minus Q so what you need to do is find out what multiple what do I need to multiply you by to get to P minus Q call that scalar multiple R and R is how we define the distance between the two points P and Q are distance R from each other where R is the scalar multiple we need to make you equal to P minus Q alright check that off there is a way to start with the unit circle and define a distance function now we get to the other side of the the story what are the possible pi values that you can get from this game so here's a number line and let's review our progress so so far we've observed that pi can equal our usual value 3.14159 and pi can equal 4 if we take the taxicab distance and with LP norms that can fill in all the numbers between those two red dots so like us to think about what other numbers do you think will be possible how high do you think pi can go how low do you think pi can go so I'll give you a hint we have still infinitely many pi values to find so this isn't all of them and I think it's good if you can invest a little bit in this experience so I'd like you to make a conjecture of how low and how high you think pi can go and there's no no pressure because you've never heard of this before so you're just taking a wild guess but I think you should tell your neighbor what your guesses are right now so take one second and do that okay so I'd love to hear some people's conjectures of how low do you think pi can go anybody provide their conjecture yes oh between three and 154 is low to high all right thank you any other any other conjectures does anyone think we can get a higher than four all right here's the answer we're ready so the lowest pi can go is three you were correct and the highest pi can go is four that was the ceiling and not only that pi is equal to three if and only if you choose as your unit circle and affine regular hexagon if you choose another shape it won't be three and pi is equal to four if and only if you choose as your unit circle a parallelogram which which worked out with our taxicab circles right they were parallelograms and our pi value was four right and if you choose any other shape for your unit circle you'll be some value in between okay all right so let's put this in a nice formal theorem so it looks it's completely clear and official if d is any norm on the plane the pi value is stuck in between the interval three and four this is proved by a polish mathematician in 1932 and and then we have another proof or another statement that the pi value is three if and only if the unit circle is affine regular hexagon regular hexagon means all the sides are equal equal length and then affine means you can you can dilate rotate or shear the hexagon and that's okay and the pi value is four if and only if the unit circle is a parallelogram and then finally you say that every number between three and four pick any number you like between three and four and there is a norm such that the pi value is that number okay now this is these are pretty strong statements if you think about it so i'm guaranteeing that if you go home and come up with your own nice distance function no matter what you do you're so constrained your pi value will not get outside this interval how could i make such a bold claim that i know no matter what you do your pi value will be between three and four i'm gonna outline how that can be true for you so that you have some intuition so let's say you go home and you decide this will be your unit circle it passes all three rules right how can i show that oh actually your pi values between three and four so here's what you do the first step is you circumscribe around that unit circle a parallelogram whose perimeter is eight now i'm not telling you how to do that i'm just saying that's the first step and it can be done and then you inscribe into your into your unit circle a hexagon whose perimeter is six and this wedges in the perimeter of your unit circle your unit circle has to have perimeter between six and eight right i wedged it in between two shapes and now if i just divide all sides of this inequality by two i see that your pi value is between three and four okay so that's definitely i jumped over how to make this blue parallelogram and how to make this green hexagon but that's the main idea of how it gets done okay back to our theorem i also think that everyone in here can can get on board with why this last statement is true pick any number between three and four how can you be sure that there's not like a skip in the numbers of pi like maybe 3.9 is not possible or something like that so it all comes from this shape here how many sides did this shape have it's a hexagon right and it's actually an affine regular hexagon so its pi value is three and watch how it morphs when i when i click through it morphs into a parallelogram okay so we can go back and forth and it can morph smoothly from one to the other so what we do is when we start with the hexagon its pi value is three and as it morphs the pi value sweeps out all the way from three and when we get to the parallelogram we know the pi value must be four because we know that if you have a parallelogram this pi value is four so this diagram shows you can sweep through all possible values of pi all right now i want to give you a big picture of what just happened i've just been slowly ploddingly introducing you to this new world of math for many of us called finite dimensional normed linear spaces and in particular you and i have been talking about two-dimensional normed linear spaces you can go into higher dimensions if you if you would like and i also want to just highlight for you that this should help you see that math is much more vast than we can ever express you spent a whole year of high school studying euclidean geometry maybe but euclidean geometry is just one geometry out of infinitely many possibilities so math is is quite vast and i also want to give you some next steps so if this was your very first time hearing about this topic and you'd like to learn more this is a very friendly book it's a paperback short friendly introduction to just taxicab distance it doesn't move on beyond that and it has lots of fun exercises and things to think about once you're once you're feeling good here if you want to be more general this is a there's an article in math horizons that i wrote called a pi day of the century every year and you can check that out it's at maybe at the college level and then this is a textbook that would be at the graduate level for this sort of thing and it it goes into higher dimensions as well all right so now i want to end with some some comments first of all you remember at the beginning of the talk i said how we celebrate pi day but now with our new perspective on the possible values of pi i just want to point out that there aren't enough days in the month of march to celebrate all the possible values of pi so you should just continue celebrating the entire month long there's no problem and also remember how we felt a little shocked when professor frink said pi is exactly three but now with our new perspective we're not quite as shocked it's possible pi could be exactly three all right thank you very much for listening all right if anybody has any questions i will bring the microphone to you you raise your hand what what can i calculate in the physical world that would be helpful to me with this abstraction with this with this abstraction abstraction okay yeah so um in the physical world maybe not so much but you can consider data so the the easy example i could give is let's make a vector and in the vector for me and one for you in the vector i have how i rated all my netflix views that i watched all the movies i've watched i have a vector and in it is my number numerical rating and then you have a vector how far apart are your tastes in mine is euclidean distance a good way to measure how far apart they are or is there something different so it turns out that mathematicians data scientists who study this they realize that sometimes euclidean distance isn't the best way because maybe the difference between your preferences in mind aren't so necessary necessarily well measured by certain movies and other movies are more important so they they use other lp norms actually to to to measure distance between data points so it's not a physical world thing but it's a real world thing hey i know you covered a lot of norms but you said everything these norms are on a plane so does this cover elliptic surfaces and hyperbolic surfaces as well if the circle wasn't elliptic surface um does it cover elliptic surfaces yeah so in general these these are geometries that you put on our end the the um and so it is not it is not considered on elliptic surfaces it's considered on um euclidean space and you put different norms on euclidean space thank you very much uh this is the best talk i've heard in moment and i go to all of them enough um so is pie changing have anything to do with time being relative no okay teleportation yeah or at least i can't comment on that maybe that's a better better response i cannot comment myself but it's a great question so let's give a hand again thank you