The Deceptive Power of Maps (with Paulina Rowinska)

Paulina Rowinska: Hello, happy to be here.

Paulina Rowinska: Yes. There’s definitely more to maps than I thought before writing the book. So, there are very different types of maps, and how I think about the map, and how also many mathematicians think about the map, is representation of reality. So, kind of the model of reality. And, it can be the map–although I don’t know how many people still have the paper, folded maps. I think this is less and less popular way to use maps. Mostly on our phones, which is also a type of a map. But then, we have, like, tube maps. I used to live in London, and without the tube map I would have been lost. So, that’s also a map, even though it doesn’t look like one.

Paulina Rowinska: Yeah. So, mapping is complicated. Why? Because the earth is not flat, and maps are flat in general. And, this is a crucial problem.

So, in mathematics, there’s a very interesting theorem that shows that when we try to take something like the earth–so a sphere–onto a flat piece of paper, there will be problems. Something will be not working, something will be distorted, be it distances or angles, and we cannot get around it. So, this is the beauty of mathematics. The theorem says so. We just can’t get around it.

And, we still are making maps, and they are still working. But, by knowing that something has to be distorted, we can be more careful when we make maps and when we look at maps.

So, for example, most of us are familiar with Mercator Projection. So this is, I think, still the most popular world map. And, well, what’s characteristic about it is the closer to the poles we get, the bigger the things look–the bigger the continents look. So, we have this massive Greenland, and then we have, like, tiny Africa. And, I think most of us are aware that this maybe is not exactly how it is in the world, really. But, honestly, I personally, even though I know this, every time I look at it, I don’t realize how huge the difference is. It really skews our reality.

And this is because of the mathematics. This is because of the way we project the globe onto a piece of paper.

So, I like to think about projections. So, that kind of translating the 3-D, three-dimensional globe onto a two-dimensional map. I like to think: Imagine you have a transparent globe with just the continents, or whatever you want to translate, drawn on them, and you put a light bulb inside of the globe. And you have a piece of paper, which will become your map. And, the shades, the shadows of the continents, will be reflected on this piece of paper.

But the question is, where do you put this piece of paper? Do you wrap it somehow around the map, like a cylinder around–

Paulina Rowinska: Or do you put it just next to it? Do you put it further away? And, this really changes dramatically what the map looks like. And, there’s really beautiful mathematics behind it. And, we can write down equations or imagine mathematically what this map will look like and what will be distorted, what will be preserved. And, again, what I want to stress is I’m not saying that one of these ways is better than the other.

I know people hate Mercator map because they claim, like, racist. Because what it shows is basically the countries that are in the north–they’re mostly richer and still more powerful–unfortunately‚ they are shown as bigger. And, the idea is that this map reinforces it. And, it might if we don’t understand that it’s the mathematics behind it. Because this map, Mercator’s map, it’s also very useful. It’s made for navigation. It’s made for navigation. So, if we use it for navigation–well, still the maps we use for navigation in everyday life–so, like, Google Maps–are based on something similar to this map.

So, again, it’s not that some of these projections are better than others. It’s just we need to understand what they distort and what they keep, and think: Why are we making the map? What do we want to get out of this map? And, choose the projection accordingly.

Paulina Rowinska: Yeah. So, Greenland is still very big. It’s the largest island in the world. Okay. Don’t get me wrong, it’s still massive. But, while it looks like the size of Africa on Mercator’s map–I don’t know the number. I wrote it in the book, I think it’s like 14 times smaller. You probably know better.

Paulina Rowinska: But, the difference is just really huge.

Paulina Rowinska: Yeah. So, honestly, I’m not exactly sure how it’s done now. But, actually, Mercator started–so the person who created Mercator’s map, he started like that. So, he started from the opposite problem. He was making globes, and how they were making it, they were making these wedges, printing–printing, drawing these wedges and pasting them onto a globe. So, it’s kind of the opposite problem.

And I think this inspired him to think in the opposite direction.

And, again, this is all math, because–it’s one of my favorite theorems of mathematics that we cannot really translate easily between dimensions, between different shapes with different curvatures. That’s the word that we should probably be using. And you know if we–if it still sounds too abstract, making maps–like, honestly, have you ever tried to put a Band-Aid on your knee? It just doesn’t stick. Have you ever seen a lollipop? It’s not–like, the wrapping has all these wrinkles, and this is the exact same reason. So, once you see it, you see it everywhere.

Paulina Rowinska: So, just, first of all, you said it’s not intuitive for non-mathematicians. It’s also not intuitive for mathematicians.

So, I think that’s the thing about mathematics. Sometimes the results are not intuitive. And, as I said, I sometimes am, like, ‘What do you mean we cannot measure a coastline?’ It just feels wrong. And so, the point we’ve–yes. When we measure a table, when we measure a straight small line, we know how to do it. Even if we don’t have a ruler, if we have a stick that we know the length of, we can just put it one after the other and approximate–it is between five sticks and six sticks–and we know more or less what it is. And, for many shapes we can do that. And, that’s kind of how we do it with coastlines. Obviously, nobody takes these days, like, a stick and goes around England, but the idea is similar.

The problem is that if the stick is very long, we are missing many of the irregularities. There are some–especially the beautiful parts of the coastline, with all the cliffs and all the inlets and bays and all these peninsulas–we are missing them. So, we take a shorter stick; and we are capturing way more, but we are still missing some. And, the shorter the stick is, the more we are capturing. And, it’s the same thing as–no. Depending on what scale of the map we are using–if we try to measure the coast of the United Kingdom using just a world map, there are not many details there. It’s not going to be very long. But, if we look at a very detailed map, large-scale map of the United Kingdom, we have way more details. So, we are capturing them, and the number grows and grows and grows. So, for a normal, regular shape, it should converge to the actual number. There is one. Here, there’s none.

Now, does it mean that all the comparing different countries and their borders or coastlines is meaningless? No, it’s not. Because if we agree on the way and the scale of the map we are using, we can make meaningful comparisons.

But unfortunately, different agencies, different countries, they use different methods. There’s no agreement. And then we can just claim whatever we want. And, I’m sure you’ve seen some people will say that the Nile is the longest. Some people say that the Amazon River is the longest. And, obviously, there are other issues. Sometimes we don’t know where the river really starts, etc. But, one of the issues is it really depends how we measure it.

Paulina Rowinska: Yes. And, what I like to think about is sometimes when you go hiking and someone asks you–like, you’re trying to estimate a distance–in a straight line? Straight-line distance is usually very short. It is like, ‘Oh, in half an hour we’ll be there.’ But then, it takes you six hours because you have to go around. Or if you’ve ever driven in a place when there are many mountains and the road really goes back and forth, you did not expect that sometimes if you use a smaller-scale map. It’s, like, ‘Okay, we are almost there.’ But, actually, you have to go back and forth. And, that’s exactly what’s happening.

Paulina Rowinska: Yeah. So, the person who actually looked it the first time, Richardson, he looked at–I think it was Portugal and Spain–the border. And he’s, like, ‘Wait a minute. Why isn’t the border the same length? It’s the same line.’ And it’s just the countries measured it differently.

And, I think we are getting better at it. It’s much harder now to find these extreme differences because just countries are agreeing a bit more, and it’s usually like one big agency. But, I still managed to find some meaningful differences.

And, just to make it clear, when we talk about differences, it’s not that the difference is, like, ‘Oh, one country is, like, a hundred kilometers more. It’s double the number, or triple the number.

Paulina Rowinska: Yeah. It’s not an error. Both numbers are correct in some way.

Paulina Rowinska: It is possible because the line drawn on the map was straight. But then, you have people–and remember, we are talking a few centuries ago, when the technology was very different. And, you had people actually going out there in the land and marking the border. And, first of all, people make errors: it’s impossible to do it in a straight line. But then, I don’t know, you have a little pond that you have to go around, or you have a little forest, whatever this is. And, these are very small differences. You really have to have a large-scale map to see them, so you have to zoom in. So, they did a really good job, actually. Like, it is approximately a rectangle, minus the curvature. But, it is interesting that something we assume–these are straight lines, that’s it. In the real world, they are not exactly straight lines. And this, we are talking about place that didn’t have many rivers or anything like that. So, imagine in a jungle, it’s impossible to draw a straight line.

Paulina Rowinska: No. No. I put a source in there. This is not my right [?]. I think I saw it in–I think it was Atlas Obscura that reported on that for the first time, so credit goes to them. But, I was like, ‘I need to put this in this book.’ Yeah. And, there is a name, and don’t ask me to pronounce this name right now, but there’s a name for this shape, which is completely useless.

Paulina Rowinska: Well done.

Paulina Rowinska: Yes. So, when the transportation started in London–public transportation in London or in any city–you have a few lines, a few stations. It’s very simple. So, you can easily put it on–you take a geographical map of London, and you put these stations there, and everyone is happy. But, the more complicated the system becomes–now London has a huge system. You can go from one side to the other; you can spend hours on the Tube. And then, it became so complicated because the lines are not straight. They just go around and turn and curve. And people were afraid to use public transport because they were, like, ‘Oh my gosh, I’m going to get lost. I don’t know how to do that.’

And, Henry Beck, who was not a mapmaker–he was, well, draftsman for some engineering company–but in his free time, he was, like, ‘What if we did with the Tube map something that engineers do with electrical circuits?’ So, a map that forgets about the geography but only focuses on the stations and the connections between them. Because, in the end of the day, that’s what we use public transportation maps for. We are not the ones driving the bus or the Tube. We just need to know which station–where to take the line, which line to take, where to get off, and that’s the important bit.

So today, the map–obviously, this is not the map that Henry Beck drew, but it’s a similar idea. And, he was not the first to come up with this idea. He just became the most famous. And, there are other cities where the maps look like that. And, now we have this iconic Tube Map with straight lines. So, they’re all vertical, horizontal, or diagonal. The distances between stations–they all look, or most of them look, they’re equal–which is not true at all.

And, I’ve made this mistake of assuming that, yeah, yeah, this is the same distance; it’s just a short walk. No. Some tube stations are just, like, three-minute walk away, and some are, like, 40-minute walk away. But they look the same. Because, otherwise it just becomes so complicated that nobody would ever use this map–because why would you? And, many, many, many cities all over the world–I live in Boston area now, and it’s the same thing here. The map is not geographical, because why would it be? It just serves a different purpose.

So, we call these maps topological. And, topology is a very weird area of mathematics, I would say. It only looks at connections within shapes, but it doesn’t really care about the exact shape. So, a circle and a square, for topologists, is the same thing, because you just can squeeze it and just elongate it. So, as long as you don’t make any holes in it, you’re fine. So, that’s why we call it a topological map, because that’s what it is. It cares about connections and order of the stations rather than the exact shapes.

Paulina Rowinska: Yes. And, that’s right, that’s what maps are. And again, I keep coming back to this in this book–why did I even write this book? Why it’s important to understand, maybe not all the details of mathematics, but the idea that every map maker makes a map for some purpose. So, we need to know what this map is useful for, what it’s not useful for, and what is there–and I think this is the most important bit–what’s missing? And again, I’m not saying that if something is missing in the map, it’s a bad thing. Often, it’s a very good thing, because we just need to have what is needed and what is crucial. And then, the map is useful. But we need to be aware of that.

Paulina Rowinska: Yeah. So, this is one of the extremely difficult problems, and one of the problems where the number of ways to do something grows very, very fast with the number of stops we are adding.

So: with five stops–if I’m a delivery driver and I have to make five stops–I have 120 ways to do that. But then, we grow very, very quickly to numbers that we cannot even imagine. And, bear in mind that every day the stops are different. So, today I order a package. Tomorrow you order a package. And they need to figure out every morning how to do it fast.

And, if you’ve ever seen a delivery driver, they are really fast. They drive with open doors, and so they jump out, throw the package, and come back in. Because that’s what they are paid for. That’s what UPS, or DHL [Dalsey, Hillblom, and Lynn], or Amazon, or whichever drivers–companies–they pride themselves in that they are fast.

So, you need something fast. That, every morning you get a route. You know exactly where to go, which way to go.

And, because there are so many ways to do that, the most powerful computer in the world will not be able to compute, every morning, all these routes and spit out the shortest one or the fastest one. Because this would take, like, centuries probably. Which doesn’t really make much sense.

Now, we also know[?] the driver to do–first, he goes to the furthest, or she goes to the furthest place, and then comes back here, and then–it wouldn’t make sense. Right? Like, if there are two stops next to each other, they probably should be served very close in time together. So, you need a route that’s good enough.

And for that, we are using different algorithms.

And, actually, they’re all secret these days. So, because this is what the company’s success depends on, really. Like, the faster they go, the more customers they can serve.

That, you know, it’s all about fuel efficiency and the time. Like, they have thousands upon thousands of delivery drivers, and they have to pay all of them, and they probably should pay more to them.

So, you know, it’s about efficiency.

So, we are creating algorithms–so, kind of ways of figuring out the fastest, the most optimal route.

Now again: we cannot really find the fastest. But we have very good algorithms that make sure that the route is at least as good as X. So, it might be, like, 10% worse than the possibly fastest one. And, it’s good enough. As long as we can get a fast answer every morning.

So, yeah. That’s, again, mathematics at its best. Also, real-life mathematics. Because in theoretical mathematics, we love the answer. We love the best answer.

But, in everyday life, we are using different ways, different algorithms, to figure out the good-enough answers. And, there are so many algorithms to do that, and they give different answers, but most of them give some answer that’s just–okay.

Paulina Rowinska: Yes. So, again, I was trying to piece together some pieces of information, I felt, because it’s all top secret. But, yes, UPS had–one day, they decided: We are going to make everything automatic. We are going to optimize that. They paid a lot of money, invested a lot of money, hired mathematicians or computer scientists to figure this out. And, this was a big failure. It just didn’t work.

And then, the drivers were, like, ‘Wait, why are you, I don’t know, making us drive at 8:00 A.M. or 9:00 A.M. in the morning next to a school when there’s no way to drive? It doesn’t make any sense.’ So, they kind of know their area better. They know which turns are more dangerous or take more time, and they just know what works, what doesn’t. But, also, we want to make it a bit more optimized. So, combining these two, the human experience–the human knowledge, and the human intuition–and the mathematical insights, and it turns out that this gives the best results.

Paulina Rowinska: Yeah. No. I don’t think even the details matter so much. Again, it’s about how–also, what are we minimizing? We are not minimizing only distance, right? We are minimizing the time, the efficiency, the cost. So, every company needs to decide what’s the priority and what’s more important. That’s exactly–also the point I’m trying to make–is that mathematics, it’s one thing on paper and one thing if we want to apply it. And, it’s okay if we take mathematics and make a bit of an art out of it. We take some results and leave other results so they suit our application.

Paulina Rowinska: First of all, when I was writing the book, I had to make so many decisions about what to leave out and what to keep in. And, honestly, it’s harder than I thought it would be, because to do that you really, really need to understand the problem. It’s the easiest to give all the details and keep the other ones.

But, I definitely like looking at, I don’t know, infographics and any data visualizations. The most powerful are the ones that have the exact amount of information that the reader needs. And, I remember listening–I think it was another podcast about–you know on the planes we have these leaflets that tell us what to do in case of an emergency. And, I didn’t realize how much effort goes into creating them. Because, you want people to be informed, but nobody will study a whole book of all the details. And, I actually–like now, looking at different planes, how differently they look, and to see what they put in, what they leave out, and what’s the most important thing. Because, again, we are humans, and there’s only as much information we can keep and process. So, the key is how to make us act.

Another example I remember: 20 years ago at this point, I visited Germany, and they are really good with recycling. But there were, like, I swear, 20 different trash bins. Brown glass, green glass, white glass. I was so confused. I came from a country that wasn’t really recycling at that point, so I was just standing in front of these trash bins and trying to figure out where trash, like, my tea bag. And, I’m pretty sure that since then Germany decided to actually reduce the number of these different bins. And to actually increase efficiency, because, if you are so confused, you’re just going to put everything into mixed recycling or mixed bag, which won’t get recycled in the end. And, I don’t know the exact details of how the recycling works there, but I think it’s the same idea: We should serve humans and not try to be the artist and try to be, like, the most efficient.

Paulina Rowinska: One thing I’ve been learning–so this year, I’ve been studying science journalism in grad school. And, writing the book, I had time. I really could take my time to go through everything and make it–it’s not perfect–but as perfect as I could make it.

Now, we were learning to write science news. And it’s a whole different story, because you have news, and in three hours or in one day maximum, you have to have a story out. And, if I started to really look into every single detail and the best way to cover the story–and this is what I was doing in the beginning, don’t get me wrong, I missed some deadlines.

So, this is another example. And both approaches are probably right, but it depends. If we just want to communicate news, we better be fast. If we want to write a beautiful feature story with interesting characters, we can take our time. But, yeah. I think good enough is good enough. And, that’s what maths and maps teach us as well.

Paulina Rowinska: So, well, it came out 2024–exactly a year ago, pretty much.

Paulina Rowinska: Yes. Yeah. It’s insane how much progress has been made in the last two or three years. I was writing the book, and without ChatGPT [Generative Pre-trained Transformer]. I don’t know how I did that.

Paulina Rowinska: Yeah. It is really crazy to see how the world has changed. And many people actually write in comments and reviews: There’s nothing about AI. And I’m like, ‘Well–‘ And, I’m not saying there was nothing happening. In fact, in my very last chapter of out–

Paulina Rowinska: Yeah. I wrote about, for example, self-driving cars–

Paulina Rowinska: Yeah. But, you know, I mean: One part was the progress is just–I think five years ago, people knew that AI is a thing, but nobody was talking about it. Now it’s like a Topic Number One. And I’m not talking about in science, but just you go on the street and people talk about AI.

And also, there are so many things I left out because it’s just one book, and I just wrote–like, maybe there will be a sequel that will be AI. Who knows?

Paulina Rowinska: Well, first of all, everything is in progress. So, whenever I hear someone making, like, a definite statement about it, I’m, like, ‘All right. I mean, I guess you know better. But I don’t know.’

My view is that I don’t think mathematics will die out as a field, but it’s going to change. And it’s changing already. So, kind of in the same way that calculators changed the field. Right?

And like, so, I think–when you think, for example, about proving a big theorem, often it’s a series of proving mini-theorems within it. So, it’s like many steps to get to this big chunk.

And, I think AI can do some of these steps, or will be able to do some of these steps. They are kind of mechanical. They involve some creativity, but they are not, like, don’t require the brightest minds.

Now, what I think AI won’t be able to do–at least for a long time–is to, first of all, figure out what the steps should be; and second, how they all fit together. And, I think this is where humans set in, to really think out of the box. To think about how to prove it, how to figure it all out.

And, I think, right now I think of AI as a very advanced calculator that can do way more than a calculator, but kind of to do the things that usually take more time than not. And, I might be wrong. And, it’s getting more and more advanced. So, I think it can prove more advanced things. It can think in a more advanced way.

But, I still think for at least some time–and a long time–we’ll need the mathematician to really think about how to approach a problem. Which problems are relevant, as well? And, what questions to ask.

So, I think still people will be in control for a while.

But, again, I might be very wrong. I also don’t work actively in mathematical research right now, so I might be missing something. But that’s my feeling. If I had to bet my money, I would say AI is very helpful. It will be very helpful. It will be more and more involved. And, I think it might become a bit like a calculator at some point. Just that the mathematician won’t spend time computing an integral, but will spend time thinking how to use this integral.

Paulina Rowinska: I think that’s a part of it. But, I also think that there is a bit–for example, this image generation, all social media people are generating images. Also, ‘Please don’t do that because this takes so much energy. This is the worst thing for the planet.’ Just saying that. And, it’s exciting. Like, I am sometimes, like, ‘Wow. It can do this, it can do that.’ And, it’s fun and exciting. But, I think–so this is like the pizza part.

But, I think there’s also–it’s really becoming a big part of our lives. Probably compare it a bit more maybe to social media, something that in the beginning was like, ‘Oh, what is this Facebook thing?’ And, now it just became, like, a thing. If you are not on social media, you don’t exist. News happens on social media and everything happens there. So, it became really just a big part of our lives. And, I think this is what’s happening with AI.

But, I also think there’s–one thing is, like, a regular user: it’s all fun, and it’s just fun and games. But then, you have the research part and the use of AI in medicine, for example. This is really saving lives–or endangering lives, depending how it’s used. So, I think it’s bigger. I think it’s bigger than everything else.

Paulina Rowinska: Yeah. And I think at some point it will become not even that exciting. It will become a new thing and people will move on to something else. But, I think it will stay. I just don’t see how we can go back.

Paulina Rowinska: Yeah. So, it is very important because the car has to–so imagine when you’re driving a car, you have to think about so many things. You have to think about the road, where you’re going. You have to think about looking. Also navigating all this equipment–brakes and everything. You have to think: Is there a child going to jump out on the road and I’m going to have to brake fast? What are the conditions? Is it raining? Do I have to slow down? Oh, there’s an ambulance coming, so I have to stop or change lane.

All these things. It’s one of the most extreme multitasking we do as human beings, I think.

And, you have a machine that has to do all that. Now, it’s very helpful if the machine at least knows the road very well and also knows what to expect so it can focus on the unexpected. Right? So, if it knows where exactly the road is, what it looks like, where the traffic lights are, etc., etc., it can focus on the child jumping out potentially. So, that’s why it’s so crucial for self-driving cars to have very, very accurate maps. By accurate, I mean it really needs to know where the curb is on the road.

Now, just a caveat–I was writing it two, three years ago. Might have changed drastically since then because the technology keeps changing. And I know that cars with AI are getting more and more–they’re more able to adjust to novelty. So they might need–I believe that in the future, they’ll be like human drivers: that they’ll be okay with going to a place they’ve never been before. But for now, self-driving cars are kind of mostly working in specific areas, specific cities, when they know the city very well.

So, what happens if you have cars that are really going road by road and mapping and taking pictures on their cameras and all this stuff to create this map? And, there’s also this kind of problem that you have, for example, a self-driving car that’s updating the map while it’s using the map to drive further. So it’s kind of like a loop. And it’s, you know, an interesting, complex problem to solve.

So, yeah: I mean, the self-driving cars are getting better and better. They used to get completely distracted by–I don’t know, if they’re driving by–there’s asphalt and then there’s grass–they are just, like, ‘Whoa. What is that?’ Now, I don’t think they are making these mistakes anymore.

So, they are getting better than human drivers. Because, people are not that great at driving, honestly. And, even though many of us claim we are the best drivers in the world. But, yeah.

So, I think that’s the importance of these maps–to kind of help the car at least know what to expect. Again, I’m not an expert there, so if I said something very wrong, please leave a comment. Yeah.

Paulina Rowinska: So, I’m not sure. I think, what I imagine–it’s a combination of better satellite images. And, so the same way that, let’s say, regular Google Maps are improving.

And also, the more cars there are, the more data is coming to improve. So, you know, if you have, like, hundreds and thousands of cars going the same road, they can be, like, ‘Oh, this map is not exactly accurate.’ Or, ‘This is the spot where often bad things happen.’ I don’t know.

So, again, I’m not exactly sure. And I would imagine that every company has slightly different ways, but I think it’s just with more data that’s coming in.

And yes, as you said: self-driving cars actually feel a bit safer. And, I think it is true also because they are much more risk averse. So, if they’re unsure, they’ll stop. They’ll slow down. While people are: ‘I’m rushing, so you know what? I am not exactly sure if I can turn here, but I’m going to do that anyway.’

So, I think that’s the kind of the benefit of them. They are–they don’t have feelings. They don’t have emotions, they don’t get angry, they don’t get tired. And, I think that’s the huge benefit of them.

Paulina Rowinska: I mean, there were a few. One is I would love to have more types of projections and kind of really look at the weird ones as well. Because there are so many weird projections that are used for one very particular thing. And, I love that. I think this is beautiful.

And, another chapter is the one that you actually mentioned about mapping epidemics or mapping crime–kind of maps that really save lives.

And, this was a funny chapter, because this is the chapter that my editor, who was always, like, ‘No, there’s too much maps.’ Like, ‘Make it a bit simpler.’ And this one, she was like, ‘Wait, why didn’t you put any maps?’ And, I was like, ‘No, this is all maps.’ And she was like, ‘Wait, but this is just, like, common sense.’ And this, I found–the best thing I’ve heard about this book.

So, I talked about how we can find things, or people, or the source of an epidemic by kind of using something that looks like common sense, but it’s actually deep mathematics.

And I loved reporting on this chapter because I had no idea about all these things about geographic profiling–so how we can narrow down the search for serial criminals.

And I had lovely interviews, and I think I would love to write more, because this is the chapter when we really talk about life-or-death situations, and I think this really brings home the message that maps and maths together are really important.

Paulina Rowinska: Thank you.