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Mapmatics: A Mathematician's Guide to Navigating the World with Maps with Paulina Rowinska
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Math & Cartography: Dr. Paula Rowinska on Map Projections, Gerrymandering & Real-World Math
In this episode of Breaking Math, host Autumn interviews Dr. Paula Rowinska about her unique journey from earning a PhD in mathematics to writing about math and cartography. They discuss the fascinating connection between map-making and mathematics, debunking misconceptions about map projections and exploring key topics like the coastline paradox, gerrymandering, and the traveling salesman problem. Learn how these mathematical concepts play a crucial role in areas like crime analysis, geopolitics, and more. Perfect for enthusiasts of math, cartography, and real-world problem-solving.
The conversation also celebrates the contributions of underrepresented women in mathematics and underscores the importance of math in everyday life. Tune in to discover how mathematics shapes our understanding of the world through cartography, topology, and even AI.
Keywords: mathematics, cartography, map projections, coastline paradox, gerrymandering, women in math, traveling salesman problem, crime analysis, topology, metric map, ai, physics, math
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Transcript
00:00:00
Speaker
Welcome to Breaking Math, where we explore the fascinating world of mathematics, mathematics, and how it shapes our everyday lives. I'm your host, Autumn Feneff. In this episode, we're thrilled to be joined by Dr. Palina Rowinska, a mathematics communicator with a PhD in mathematics.
00:00:18
Speaker
You may know her from her well-known TED Talk, Let's Have a Maths Party, where she brilliantly highlights how mathematics is woven into everything around us. Paulina is also the recipient of the Imperial College President's Award for Excellence in Societal Engagement.
00:00:35
Speaker
And she's also a creator of interactive content on brilliant dot.org. She's the author of Mapmatics, a mathematician's guide to navigating the world. And today, she's here to deep dive into the beauty of maps and math with us.
00:00:57
Speaker
How are you doing this morning? I'm doing well, thanks. How are you doing? I'm pretty good. It's actually really nice. We're both in Massachusetts. So yes, beautiful Saturday as we're sitting here and recording. So we're really lucky. ah Now tell us a little bit about yourself and what inspired you to write mathematics. So I have a PhD in math. ah And while doing the PhD,
00:01:30
Speaker
um I just realized I prefer to talk about math and write about math and science rather than actually do it. So I was just like running from one conference to another, from one talk to another. My supervisor was always worried I won't finish my PhD in time because of that. ah but ah But I did. So ah and after that, I Well, I wanted to write, but I got a bit scared of the like job insecurity in um writing and journalism. So I did what many math people do. I went to the industry, and which was a big mistake I did not fit in at all. ah So then I started writing and ah I had a full-time writing job.
00:02:24
Speaker
and on top of that I decided to write a book which had been my dream since I was a kid really so that was kind of the the reason I wrote it. I wrote i wanted to write about a math for people who are scared of math because let's face it many people had not had the best of experiences at school and I love maps and I was like there must be a book about math and maps but there wasn't like I couldn't find anything anything so I wrote it ah so this is the long story short and right now I'm here at MIT doing a science writing program so back to school after a couple of years sometimes that happens I know the feeling when you're going from
00:03:17
Speaker
one career shift to the next. And I think a lot of folks in mathematics end up in finance in one way or another. I ended up ah in crypto and blockchain after. Yeah, makes sense. That's where the jobs are, or where the money is. And you know, these jobs can be very exciting, but it takes the right person.
00:03:41
Speaker
It does, absolutely. Now, how did you become interested in the intersection of mathematics and cartography? I always loved geography. Like in high school, geography was my favorite subject. I did the National Geography Olympiad. I don't know if it's a thing in the US, but it was in Poland. We have some of them like ah Reader's Digest, National Vocabulary B, they have a geography section. I know that I had that when I was younger. Yeah, so I basically spent my ah last year of high school studying geography. um
00:04:24
Speaker
And then I kind of didn't do anything with it, but it always stayed with me. so you know, it's like, it's hard to write a book about math. ah And you want people to see that math is actually used somewhere that is actually important. And they say write about things you know, and things you love. And that's a very good advice. Because yeah ah I spent years working on this book, and I never got bored. um So
00:04:56
Speaker
Yes, I said I just couldn't find a book about my two passions. And that's how it happened. Like, there's no bigger story behind it, really. Like, I just thought I would write something about my hobbies. And ah the intersection is actually big. Like, yeah, well, I wrote the whole book about it. And I could write probably a couple of other books about it, because I had to cut out so many things. Who did you write the book for? that that's That's where I would like to know because it's very relatable, understandable. um so
00:05:29
Speaker
i wrote this book for people who are non-scientists, like you don't have to be a scientist, especially not a mathematician or a cartographer to understand it, although I still hope that some mathematicians and cartographers will pick it up. um But so I assumed like, you know, you finished math education, like high school so you don't need to know really very much to read this book and I wrote it for people to show them that math is interesting and also for some people to show that maps are interesting because ah
00:06:09
Speaker
you know having this intersection is maybe someone will pick it up because they're interested in maps and they learn something about math and the other way around and the book is mostly stories so it's not a textbook you want you know you won't pass your exam based on this book but ah hopefully you'll get interested and maybe you'll you know read something more if you're if you want to know more. So yeah, it's a typical popular science book, and not a textbook. Now, when you were laying out the book, what are some common mathematical misconceptions that people have about map making? Maybe we can dive a little bit into topology and talk a little bit about curvature as well.
00:06:57
Speaker
Well, first misconception is that people don't realize that there's any math in maps, apart from like some numbers that appear on maps. I'm not really talking about these numbers. To make a map, we need some serious mathematical um knowledge, or at least Maybe not knowledge, but intuition, at least. ah so Well, I'll start from the curvature, because what happens when we make a flat map? Well, we take our globe that's three-dimensional. I mean, in it's it's a sphere, so like it's ah it's a surface of a three d um-dimensional object, but it's not flat. And we need to flatten it.
00:07:42
Speaker
So what we know from Gauss already which was ah centuries ago a very eccentric mathematician and not only mathematician, he figured out that it's actually impossible to flatten a sphere without making any, ah let's call them mistakes, many any um anything that these distortions might happen. And how can you see it actually in daily life? Well, for example, if you have if you peel an orange, you cannot really flatten the orange peel ah without tearing it or or just distorting it in some way. Or ah
00:08:27
Speaker
you know, like when you recently I fell down while running, and I had to put a lot of bandaids on my knee. Well, it's really hard to put bandaids on your knee, because the knee is ah round, let's say, and the bandaid is flat. So there's always like this little pieces that just don't fit. ah So you can and this is the same principle for maps. yeah So ah and and it's actually you can prove it. ah This is because of the curvature, they have different curvatures. ah so um and And you don't really need to know what curvature really is. It's a mathematical concept. But intuitively, like this sphere is curved, the flat piece of paper is not. So because of that, something will be distorted. And it's the job of the map maker, the cartographer, to decide what will be distorted.
00:09:19
Speaker
So for example, some maps distort the shapes, some maps distort the relative sizes, ah the lengths, so the angles. And so you have to decide what's really important for you to keep. And ah this decision will impact how you make your map. ah And we have many different types of maps. We call them projections actually. So how we like project this three dimensional object on a two dimensional space.
00:09:49
Speaker
And ah that's why we have all these different maps. ah Yeah, so it's ah it's a really important mathematical concept that really, really, I would say it's the key decision when we are making a map of a large space. If you're making a map of like a city, it won't matter much because ah when you look at the globe, like the city is basically flat, approximately flat. ah But when you make a map of the world,
00:10:18
Speaker
then you really have to take some serious decisions. Absolutely. Now historically there was some competition between nations on what or who was right. So when we're delving into topology and building out these spaces for the projections of the map, tell us a little bit about the history of it.
00:10:44
Speaker
Today we know, most people know that the Earth is almost a sphere, but it's a bit flattened at the poles, so it kind of had this has this like grapefruit shape. And we knew, so there's that actually a common misconception that like we thought that Earth was flat until Great Columbus came, which is not true. like In general, we knew that the Earth was not flat since ancient times.
00:11:10
Speaker
but ah people assumed it was a a sphere and and at some point they realized ah that it's actually not exactly a sphere ah but then we had two nations that didn't like each other very much which was ah well it was England and France and ah one of the countries claimed that the Earth is flattened at the poles so it resembles a grapefruit and the other days actually elongated ah so it's kind of more like an egg and oh they could not really agree what it is and you know you would think it's like a silly argument like why why would you argue about it but you have to bear in mind that
00:12:01
Speaker
Like up to quite recently, ah most of the travel ah happened on the sea. and They were going to the lands and claiming them for themselves. These were colonial times. and Not a great history, but that that that's what it was. So ah understanding the shape of the earth was really important to be better at navigating the sea.
00:12:28
Speaker
be faster and more efficient in like grabbing these new lands and also it's you know after two days like who will get first to the moon who will do first this or that like every nation wants to be first because then they feel like more powerful um So France sent out, ah France and Spain, and it was like, ah that a I write a bit more in the book about it, but there are whole books about just this expedition, yeah because ah it's it's just a fascinating piece of history. So they went to ah South America around the equator and to measure
00:13:10
Speaker
ah really measure ah by hand the length of one ah degree of, ah now I'm gonna get it wrong, longitude. ah So to kind of understand if it's ah longer or shorter than the one closer to the pole. And this but this would show what's the shape of the earth.
00:13:36
Speaker
Now, this was supposed to be a short trip. It took decades. ah Because you took these scientists who were used to just like calculating stuff and ah pen with pen and paper and they actually had to go into the mountains. There was snow, there was heat in some areas. There was, ah you know, there was a jungle, there was like,
00:14:03
Speaker
they were just not ready for that. and Meanwhile, you know, some people ah like, had some interesting ah ideas of like, giving all the money to their lover of all the expedition money. It's a crazy story. ah But in the end, they figured out that yes, the earth is ah has grapefruit shape, ah but it took decades.
00:14:29
Speaker
I have to say that that's kind of ah funny hearing the story for behind the scenes because, you know, as mathematicians, you don't usually ah mix the theorists with the people who do the applied work.
00:14:44
Speaker
yeah yeah Yeah, and I think this is a today thing, yeah because we are really specializing now. like when When you read about i don't know Gauss, he was doing everything, he was doing experiments on himself, and he was also doing theoretical mathematics. So ah it really changed, I would say, in the last century.
00:15:05
Speaker
absolutely it's kind of just that whole ah shift of what are you specializing in? And some people are really good at that one particular thing, but then, uh, it's, you don't put somebody who's a, how do I put it? A theoretical, like chemist into the lab. Yeah. Yeah. It might not done very well. Yeah. So, you know, those things happen over time. Now with that, uh,
00:15:37
Speaker
If you're talking about the different types of maps and if we're going to look into a population, what they use, ah
00:15:51
Speaker
I'm going through this. Hold on. so If we're looking at like the current maps and current projections, could you tell us a little bit more about the coastline paradox? Because you talked a little bit about longitude and latitude, and how does that work? Right. So when we have our map already, and we are looking at, it it has some lines. It will have coasts. It will it will have borders.
00:16:26
Speaker
and it's one of my favorite math facts is that ah like lines are essentially infinite or they can have any length we want.
00:16:41
Speaker
And how does that work? Well, it depends on the scale of the map. So how many details we put on the map. So if it's just like a, you know, word map and you see a river, it's like basically a straight line with a few little curves. So you can take a ruler, measure like one piece of it and measure another, and you get some approximation of that length.
00:17:10
Speaker
Then if you look, let's say, at a map of Europe and you look at the same river, there will be a few more details. So you can take like ah you can just approximate it a bit better with this little these additional curves and ah lines.
00:17:30
Speaker
Then you look at the map of, I don't know, Poland and you look at the same river and you'll have more details and you can keep going. And the more details you add, the longer.
00:17:42
Speaker
the ah line will be. This is only about lines that are, ah well, any natural lines, I would say. So if you have a straight line, it's going to stay stay straight and the the length won't change much. But if you're looking at the real world lines, ah like coastlines for example, the length might change a lot especially in places like I don't know the coast of Norway that they have these fjords ah that can go for kilometers into the land and then really the scale of the map like can change it by a factor of 10 or 100 or um yeah so be really careful when you look at any
00:18:28
Speaker
ah ranking of the longest borders, longest rivers, longest XYZ. like it's um It's really misleading because you really need to make sure they are comparing apples to apples, not apples to oranges. So they were all measured on the same map.
00:18:49
Speaker
um And I find it really interesting because we look at like, okay, well, how how difficult is this to measure a line? Well, not not very difficult, right? It's the 21st century, like we can do, you know, we can chat to our computer. And yeah, but we cannot really measure a line I mean, we we can, but it's always an approximation. And it has some reward consequences. ah You know, like, in some states in the US, for example, the funding and for coastline protection um depends partially on how long the coastline is. So, if you report the
00:19:33
Speaker
coastline lengths on different scales, some ah states will be privileged to you know in with respect to others. And yeah, so they are all just like political and ah financial consequences. Yes. And also, a lot of the places in the US, s the coastline is actually shrinking.
00:19:57
Speaker
And you know you're currently in Boston. I remember a few, even about 10 years ago, some of the harbors and marinas, the water level is almost at the top of where some of the docks are. like r So if you go down to like the piers in the Seaport area before, that coastline was way further out a few years ago.
00:20:25
Speaker
And now it's also coming up. So we only have that approximation ah for where a lot of these lines are. Now, what do people use for our regular maps on day to day? We have the, what it's called Mercator map. And that's what the majority of the population uses. So how did that come about? And why do so many people know that as the standard?
00:20:52
Speaker
Yeah, Mercator Map is one of the most famous projections ah and well, it's really old. It was ah created centuries ago and and it's still popular. It's actually used a version of it, quite a bit modified, but it's used by online mapping ah services, ah which is why we are so used to seeing it. I was taught on MercatorMap my geography at school and many people were. And ah so MercatorMap keeps the angles. So you know how I said that some things have to be distorted when we ah project ah the globe, some things stay ah the same. And here the angles stay the same.
00:21:43
Speaker
um So the shapes of the lands are perfectly fine. It's what you would see ah if you, you know, went into space and saw the Earth.
00:21:54
Speaker
um Now, this was important because this map was made for navigation. And when you are navigating, then ah the angles are what really matters. But what ah this map distorts are the ah sizes, relative sizes. So I grew up looking at this map and I still have in my brain that Europe is big.
00:22:24
Speaker
And it's it's my brain, like Europe, is big maybe like half the size of Africa, which is absolutely not true. Like Africa, South America are massive. ah But we don't fully realize that by looking at this map where Greenland is like the size of Africa, basically. yes so um yeah Because what this map does, it's makes it makes everything close to the poles bigger and everything close to the equator smaller. And and this is really powerful, especially given that ah what's close to the poles usually is richer, more more powerful regions, ah while what's around the equator are in general countries that are still climbing to to get
00:23:13
Speaker
more powerful. And we like this narrative, this map really reinforces this narrative, which is not great. um On the other hand, many people really hate Mark Hader for that. And I would argue that I'm not saying he was not racist, and he like cared about people in this other like, let's say in Africa, South America,
00:23:37
Speaker
But I also think he had good reasons to make them up ah the way he did in particular navigation. He made it for navigation. Also, we didn't know he didn't know very much about ah anything that wasn't like Europe, North Africa, because we haven't traveled really ah that much around. these were These were the times of the discoveries, so discoveries European discoveries.
00:24:07
Speaker
and because of that he could kind of hide his lack of knowledge because Africa was so small so like the lack of details wasn't that striking. Well he could put all the places he knew on Europe so this was quite convenient as well ah but today the map is well quite problematic in a way if it's the only map we see especially as children then we build a wrong view of the world um in our heads. ah But I'm not advocating we don't see it because I really think we should see it and ah but with the understanding what it means and what it represents.
00:24:51
Speaker
I think ah you did bring up a really good point with that because oftentimes the way that Europe views nations versus living in America is completely different. We don't always think about the ah like the power of how one culture influences like the rest of the world.
00:25:15
Speaker
I think that that's something that people don't usually talk about as much, especially with you know the smaller countries that are on the map don't ah usually have more power in one way or the other than a lot more around the equator.
00:25:35
Speaker
Yeah, and yeah, we just, I mean, I think here we are like, we can really talk for hours about it, like how we are taught at school. ah Yes, how much attention is put, you know, at school, we learned a lot about Europe, which is natural, I grew up in Europe. So this kind of makes sense. But also a lot about North America, a few Asian countries and the rest is like, yeah, there are some countries there, ah which is ah You know, not the fault of the map at all. It's just a narrative we're building yeah and ah hopefully this is changing. I really hope it is. Oh, oh we have ah different sorts of maps, right? Beyond the Mercator map and some are more topological use and others are metric maps. Can you talk about that and the difference between these maps?
00:26:34
Speaker
Yeah, so this is mostly about, ah let's let's zoom in and let's look at like a city, sure so a much smaller area. ah And it's one of the like most interesting bits of the book to write was ah about how we ask like as individuals navigate spaces, like cities or even you know a campus or a building.
00:27:02
Speaker
um because this is really interesting how our brains work. And this also can be seen in different maps of local areas. For example, ah like and the iconic map the of London Tube, so London Underground, is ah I think anyone who has ever been in London, they will remember this map because ah it's ah it has many straight colorful lines ah but when you put it on the let's say a real map of London it doesn't make any sense ah because what this map does is ah it's a topological map, a topology
00:27:50
Speaker
ah It's a field of mathematics like mathematics that I would say studies connections and ah it's not really concerned at all about um the distances. ah And this is what this map does. So it ah you when you look at the map, you can see which stations are on which ah tube line and which order the stations are. ah So you can figure out how to get from point A to point B. But you have no idea how long it's going to take.
00:28:26
Speaker
and so never ever use tube map to walk around London or in in fact any other city I would say ah because two stations ah two pairs of stations that seem to be at the same um ah distance from each other one can be three minute walk one can be 45 minute walk And it happens to me when I just moved to London, and I didn't even think about it. I had this tube map and I like i didn't have an internet on my phone yet. And I spent like a long time walking to a place. I got there finally, but... um
00:29:07
Speaker
Yeah, ah so don't do that. Be smarter than that. and but But these maps are extremely useful because ah if you've ever seen like so so some people have made like a geographical map of London transportation system and other in other cities as well.
00:29:26
Speaker
and it's just too much detail, it's too confusing um because you just want to get on the tube and get out at some station and you don't really care if the ah the train is turning or not, if it's ah like it it doesn't matter because you you have no impact over it.
00:29:46
Speaker
However, when you are walking around or driving around, you might want a metric map. So not a topological map, but a metric map. And the metric map is, ah the distances are important there. And ah that's what you want because you want to know exactly how long it's going to take you. um So these two types of maps are both super useful, but for different applications.
00:30:14
Speaker
Wonderful. Now, when we're looking at a different application, let's let's go into graph theory. So say we have our, ah whether it's an Amazon truck or, you know, you're trying to optimize some sort of transportation system. How you go into a little bit of the traveling salesman problem in the book. ah So tell us what that is and how that ah plays a part in maps.
00:30:45
Speaker
The traveling salesman problem is, ah it sounds simple. So let's say you're an Amazon delivery person and you have to deliver ah packages to 10 places today. So you have the addresses, you can put them on the map.
00:31:05
Speaker
and you need to visit them all once ah to drop off the packages but how do you design the route that's optimal so that's ah the fastest and here the fastest can mean the time or can mean the distance ah this is up to you to decide um well it turns out it's extremely difficult to figure out because ah you can figure it out maybe with five maybe with 10 places you still can, but if it raises to 20, 50, 100,000, like, there's no way not even like the most powerful computer ah will be able to figure it out. And why is that? Because there are so many possible routes between ah these places that um you just cannot compute them all.
00:31:58
Speaker
ah So the problem is simple, but the solution is just impossible on practical level. So what we do instead is we create different algorithms that approximate this solution. So you will not get the best possible path you can take, but you'll get something that's just all right. So something that won't be super long, but will be ah short enough ah for you to be quite efficient. ah Because otherwise, you know, these decisions have to be taken fast. So every morning, ah every delivery driver gets a list of places because it's also not that the places are the same every day, right? Like one day I order a package, another day you order the package, and they have to like go to different places.
00:32:49
Speaker
So one cannot spend ages, like it has to be fast. So that's why we have these different smart ways of figuring out. They are very different for different deliver delivery companies. And they are really kept top secret, ah because that's what the company's success ah relies upon. that There are some estimates in like one minute when a driver is like one minute of loss or lost time costs like some crazy amount of money. Yeah, ah which I did not realize. So actually, that's why you see often drivers that have like open doors, they drive with like cars with open doors, because they like want to jump out and run back in. ah Which doesn't seem safe for me, but that's a whole different story.
00:33:34
Speaker
um ye So yes, so we have, i I talk in the book about a few ways of approximating this solution. There are so many, there's no way I could talk about them all, but they are also so different. Like my favorite, I think, is that ah the one that ah kind of takes ideas from the animal world.
00:34:01
Speaker
so you know how ants they always find very efficiently a way to any cramps or any food you leave around ah and they always like go they are all going in the same way but how do they figure out the way actually they're really good at it so they first go in random directions ah to find food, but then whichever ant gets the food, ah so finds the food, they bring the food back and ah then they leave pheromones, so these substances that you would need to ask a biologist what exactly they do, but in in short they ah change their behavior of other ants.
00:34:47
Speaker
and then ah the more ants come back on the same ah path and this pheromone track is stronger so then the other ants figure out like oh let's go there because there was food and this way they strengthen some paths and make other paths ah just no ants will go there and that's what we do in this algorithm like obviously the instead of ants we have um ants let's say are coded up in a in a program. ah But I really like it because it's like, oh, we think we're so smart, but answer smarter. So we takes ah their ideas and we implement them. um So it's just one example. Wonderful.
00:35:31
Speaker
Now, we know that the optimization and scheduling has isn't really a new problem. But one big problem that I noticed in here was how how does math and maps tell us more about crimes and understanding crimes? I found that to be really interesting.
00:35:53
Speaker
Yeah, um so it's also one of the most interesting interviews I've done for the book. I spoke to ah Dr. Kim Rosmo, who's a police officer, was the first police officer to in the u to in Canada, sorry, to get a PhD,
00:36:12
Speaker
ah ah if I remember correctly. Very interesting, actually. Yeah, so he came up with and with this formula, this method of tracking down syria a killer serial serial criminals, so also ah burglars, rapists, all terrible things.
00:36:41
Speaker
So um serial crimes are very different to your everyday crimes because often the criminal, they they choose their victims ah not based on who the victims are, they don't know them personally very often.
00:37:01
Speaker
it's just a victim is in the wrong place at the wrong time. And that ah that's, how that's how it in general works. There's a whole psychology behind it. yeah um So any, and actually,
00:37:20
Speaker
Finding them is, I mean, it's very hard, but there are some ah interesting bits we know that can help us. So what we know is actually serial killers, and ah they they like to stay in their neighborhood of their home or workplace, ah so they usually don't venture out really far away.
00:37:43
Speaker
However, they will not find their victims super close to their home, um possibly because it might be, you know, it will be too easy to catch them. So their the probability of finding ah the base of the serial criminal is actually if you kind of draw it on a map and if you assume that where the high probability is like a high point over the map and low probability is like a low point, it looks a bit like a volcano.
00:38:16
Speaker
So in the middle you have like a big hole ah where it's quite unlikely to find them. Then you have very high probability and then it kind of goes down the further away we go.
00:38:27
Speaker
so ah So what now police officers can do is they ah look at different places where the crime, basically different places where the criminal has been. It can be where the crimes happened, it can be, I don't know, where some ah DNA traces were found, anything they can figure out. And they ah use these points to calculate probabilities of different points on the map. And this is usually a local map of a county of a city ah to figure out these probabilities. And thanks to that, it will not let them find the criminal. out But what it will let them do is to narrow down um their search area and prioritize the places with high probability so that they just have higher chances of of finding this person.
00:39:21
Speaker
And it's worked. It's worked many times. ah This was Kim Rosmos. It was his PhD thesis and actually like it was applied and they managed to narrow down. It was a case like decades ago at this point.
00:39:39
Speaker
and they found the person. also I wouldn't say ah only thanks to this method because there was much more work going into it, but of course it really helped. So this is how like math and math ah ah math and maps intersect to to really save lives, I would say. That's really, really interesting. Now, if we're looking a little bit of a topic shift here,
00:40:08
Speaker
Another thing that you talk about is gerrymandering in maps. Tell us a little bit about that. I know in previous episodes we've had people talk about mathematics and democracy, but we've never really gone into just the map perspective of it.
00:40:27
Speaker
Yeah, gerrymandering is really important right now when this episode will be out just before the elections. ah And gerrymandering, what it is, is and drawing the maps the electoral maps in a way that are biased towards one party.
00:40:49
Speaker
and it's it's it happens everywhere in the world, but in the US it's a huge problem because of the voting system here. ah So how it works here, as most listeners will know, but I'll just ah remind you, so it's basically there are two parties and so ah in each electoral um district and when people vote there's just one winner and this one winner ah so for example if the electoral district is a state ah each state brings one candidate and this ah this is you know kind of tricky because states have different numbers of people so not all votes count the same
00:41:40
Speaker
um but also on a smaller level when we draw these districts, they can be drawn in a way to kind of squeeze all supporters of one party in like just one district or a small number of districts. So they win these districts. Absolutely they do. The problem is that they lose all the other districts. So like all their votes are kind of wasted to go into this one district.
00:42:07
Speaker
And there i mean there are many battles of doing this gerrymandering, especially these days when we have access to ah really, really detailed data, ah politicians know exactly where people live, who's voting for whom, ah it's really complicated. And it's really hard to detect where when a map was gerrymandered, because just looking at the shape won't tell us much.
00:42:32
Speaker
And so it's a big, big political problem. Many, ah many scientists are working on solving it on ah different ways of detecting it. um And it's really a collaboration between politicians, political scientists, mathematicians, sociologists, computer scientists, like you have to have a really multi multidisciplinary teams. And Yeah, there are there are many aspects of it. Like what I found really striking is how we gerrymander indigenous communities or people of color or just minorities that are usually underrepresented and they really need to be represented. Yes. Oh,
00:43:19
Speaker
in on different levels of the government ah and it's so easy to pack them in one place and claim they like oh they live there anyway so ah they live there together the anyway so like you know we did nothing wrong.
00:43:36
Speaker
But it's way more complicated than that. So, I mean, there again, this is one of these topics, their whole books about gesture and mandarin. And I had only one chapter. ah But the topic is really um important. And again, this is, again, we have maps and math working together to either do something bad or something good, yeah depending on who's using them.
00:44:02
Speaker
Exactly. It's like you need to shift, whether it's a zone or whatever, because your population's grown too large. yeah And that can change a whole vote for a whole state.
00:44:16
Speaker
Yeah, yeah, you absolutely can. and and And people are doing stuff about it, not only scientists, but like in some states, at least the and new maps are drawn by um independent ah people, and not politicians themselves. ah we I mean, it's hard to stay unbiased if you know that you are going to be one voted for or against. I'm not even blaming them exactly. like I don't know how I would do it the to be unbiased. As I said, it happens all over the world, but in in some in in many countries, the system is different. So
00:45:00
Speaker
It's not that you get one candidate from every electoral ah district, but it depends on, for example, you count votes folds from the whole country and you like rank the candidates. Rank choice is indeed better. It's mathematically proven. but yeah Yeah, but, you know, that their argument, and you know, as a mathematician, I definitely don't have the expertise to really ah say what's best, what's not, because, you know, we can prove mathematically that something is better. But then there are all other aspects, for example, sometimes we gerrymander on purpose, and it's a good thing to actually give
00:45:42
Speaker
these minorities underrepresented minorities a chance. And that's the good gerrymandering, I would say. So there's, ah you know, this is the where it's not like some people think that math, you have an equation, you solve it, and you have one answer. No, here's the human aspect and the math can guide you. Correct. But you as a a human or, you know, a lawyer or a politician, you have to decide what you do with this information. And there's usually not one single answer. And that's, that's why I think we need to collaborate between between sciences.
00:46:20
Speaker
Yes. Now you you talk a little bit about underrepresented groups and having them represented. I know that in the book you do highlight ah a few particular women that are ah less known. Can you tell us a bit about one or two of your favorite people that you talked about?
00:46:45
Speaker
Yeah, so in the last chapter I actually, and this was not on purpose, I wanted to write about mapping and the bottom of the sea floor and ah the inside of the earth. And both of these ah types of maps were actually that developed like the the big characters in developing these two were two women that I must admit I haven't hadn't heard about them before and they are extremely important for science. So one was Marie Tharp who
00:47:24
Speaker
ah Basically discovered or was one of the people who really contributed to discovering how tectonic plates work And this was so recent. This was I believed in the 50s 60s and ah And I somehow assumed we had known it for a long time But actually it seems that like my parents were the first generation to actually learn about it at school ah which is so pretty crazy for me like for me the tectonics were just like something we had known forever um and ah so what she did is she was a woman so she couldn't go ah on the for the expeditions to measure the depth of the seafloor she had to stay on land and the men were bringing her their results of ah measuring the depth
00:48:24
Speaker
and she was supposed to analyze them and map them. And she came up with this amazing idea of mapping the depth ah depth of the ocean ah on like a map of the ocean itself. So we could kind of see um where did where the ah different measurements were taken.
00:48:47
Speaker
And she was like, wait a second, there's something weird going on there. And she discovered basically the Rift Valley. And that we know now is this huge rift going through like whole Atlantic basically. And ah I mean, the Rift Valley is on land, but it's just a big rift. And, and then she also talked to a colleague and he looked at it and he was like, oh, but there's where the earthquakes happen. And now we know that earthquakes are really ah where the tectonic plates meet. So people were not believing them, like she she's just a woman, so what can she know? and ah And it really took a lot of evidence and persuading and a lot of this
00:49:35
Speaker
ah beautiful maps that most of you will have seen this National Geographic map with this depth of the ocean there. yeah And that was made in collaboration with her. Amazing. Yeah, so she sets um and she said that you know like that the pictures, the images are more important than words very often because that's whats what's true. And um I'm really surprised that we she's not like the household name, ah like Einstein is, and she should be. And another woman ah was Inge Lehmann, if I'm pronouncing correctly, I don't know. ah She lived in Denmark, ah and she made a huge progress in mapping the ah inside of the Earth,
00:50:29
Speaker
So the different layers of the Earth, ah which is actually really hard to map because we cannot just like look at it. So we have different ways, but ah the majority comes from actually earthquakes that happen somewhere in the world and we measure when the seismic waves ah reach the observatory and based on that we can figure out like um what they went through so we can kind of understand what's inside. ye And ah she did amazing work in that. ah
00:51:07
Speaker
On one hand, she was responsible for the some of the observatories and she really improved the quality of data coming in, which was very important, but also she looked at the models of the Earth that were created before.
00:51:24
Speaker
And she understood that there's something wrong there. And she did something that ah I really loved. She simplified the problem. ah All her colleagues, all people so far were like put ah writing all these complicated mathematical equations that they couldn't solve. they And she was like, let let let let's forget about all that. Let's make like the simplest model pop possible and take it from there.
00:51:51
Speaker
And this was the key. This simplification really helped her understand what was going on. So obviously since her many people have improved the maps of the inside of the earth with have better equipment, with ah better computers, but she she made this huge jump. ah And that's how we know what's underneath our feet really. Wonderful.
00:52:14
Speaker
Now, when you were going through all of these other stories, was there anything else that really surprised you when you were ah trying to pick the topics in the book? Oh, there are so many things I've learned, ah like some I've already mentioned, some, yeah, i for example, like the talking about the brain and ah there's this huge debate on whether we have metrics or topological maps in our in our heads. So do we navigate by like understanding which way to turn and
00:52:49
Speaker
ah ah just like looking at, yeah, I have to turn here and then I have to turn there. Or do we navigate by kind of understanding vaguely where we are in the, what's the bigger picture? So which, which direction we should go to. And turns out we still don't know. And um that was fascinating for me.
00:53:12
Speaker
I really loved learning about the whole history of the four-color theorem and its proof. um So four-color theorem is like super simple a theorem that ah every map, like political map, let's say, can be colored with just four colors ah in a way that and no two touching countries have the same color. Super simple. You can give your kid, ah if you have a child around, ah you know, ah some markers and a map and they can play around with it. But proving it for every single map that can be ever created is not that easy. And it took centuries and they're reading about the story of it and understanding the the history of what, of this all this wrong, proves what went wrong. And this was actually the first ever
00:54:08
Speaker
ah computer-assisted proof. yes um And ah it was very controversial. Today we are rather accepting ah computer assistance with mathematical proofs, but back then it was something new and really frowned upon. And ah yeah, like again, the the story was just yeah And understanding the proof took me a while. like I remember reading about it. It's not easy to understand what's going on. And actually, this was the chapter I had to rewrite so many times because ah my editor was like, i I don't get it. And I looked at this, yeah, probably I complicated it a bit too much. So ah which was a good exercise also for me to understand what's actually going on. So I also learned while writing this book, not only learn to write, but also learn the actual math. I think I learned that originally when I was doing origami. And you can map out the sections and the folds using ah whether it was three colors or four colors and certain theorems and the mechanics just fold really flat with that. Yeah. um So same map, same application. Yes.
00:55:31
Speaker
um now Was there anything that you wanted the listeners to take away from this conversation today? I would say Well, one thing is that math and maps really go together in both ways. And it's really important to understand some basic of the basics of the math, ah to just not be fooled by maps. And we are surrounded by maps, you know, in the news, on the internet, like now there are unfortunately many wars going on. We are seeing all these maps, there are elections going on, we are stealing all these maps. And the map maker,
00:56:10
Speaker
will show you something they want you to see. So you need to understand how they, not super detailed on a the level, it's not your job. But your job is to understand like, what projection is it? Why is it this projection? What are they trying to hide? Or what are they trying to highlight? So it's it's it's really important ah to to get it. And um another thing I would say is that You know, don't be scared of math, ah because you don't need to do equations to do math. you ah
00:56:48
Speaker
I describe in a book like when we are looking for lost headphones, I think like you're doing math, you just don't know you're doing math. Exactly. And ah that's something I wish we were taught at school. And ah whenever I mentioned what kind of book I wrote, I get this like scared looks, oh, I'm sure it's interesting, but not for me. And I'm like, no, it is for you. ah If you have any interest in um not even learning, just reading some stories and maybe learning something as you go. So yeah, these are my takeaways, I would say. And yeah, please do read my book if you like it, let me know if you don't like it, definitely let me know. ah I would love to hear, you know, your feedback as well.
00:57:35
Speaker
So thank you very much for coming on Breaking Math. Thank you. For listeners, if you're interested in the book, mathematics is on sale in the United States as of September 15th, 2024.