When I was a kid, every time we had oranges, my dad would always peel this little orange man into the skin as he was cutting them for us. Which was quite fun, really. I mean, like, get to have like a little companion. (Brady: Why would he do that? Was it to encourage you to eat, or just because he liked entertaining?) You know what? I'm actually, I'm actually allergic to oranges as well so it'd be all nice and fun until I was sick. (Brady: Your dad suddenly doesn't seem so nice!) But the mathematical idea of peeling a two-dimensional object into the surface of a sphere has stuck with me, has stuck with me. Oh hang on, I've lost my place now. Okay we're on, we're head, arm, leg, leg - we need another arm. The reason I wanted to show you this was, because there was always something a little bit annoying about the orange man- (Brady: Oh no!)
- It's fine. His legs coming off now. Wait, wait - imagine you didn't see that bit. Okay, no, it kind of works. It kind of works right? It kind of works. (Brady: Let's see him)
- There you go. He's sort of there - I chopped his leg off. (Brady: That's terrible!) If you- well, whatever. He's kind of cool, like this little orange guy, he's more like a starfish really. (Brady: Here's one Hannah made earlier) Okay, my early one was a bit better, my practice one was a bit better. This is a fresher orange, I think that's my problem. (Brady: I only just realised that your earlier one has got this-)
- I know exactly! [laughs] The problem is though, even though you've got this kind of quite fun little guy, you can never get him to sit flat on the desk. You can never quite get him to, like, lay perfectly flat. Especially not without really, you know, deforming the orange peel. And there's a reason for that. And the reason is something called the Gaussian curvature. Now, Gaussian curvature is a number that describes how curvy a surface is, and it's made up of two things multiplied together. One is how the surface curves in one direction, and the second number is how it curves in the perpendicular direction. You multiply those two together to get the Gaussian curvature. No matter how you bend a surface the Gaussian curvature will always stay the same. So here with an orange, it's positive in that direction, positive in that direction, so whatever you do to the surface of an orange it always has to have some kind of a bend away from you. You actually want your orange man to lie flat on a two-dimensional surface which has zero Gaussian curvature. So there's no way that two positive numbers can ever multiply together to make zero. And that doesn't really matter so much when you're talking about orange peels or little orange men, but it does matter when you really want to make the surface of a sphere lay flat on a two-dimensional surface. Like when you're trying to make a map of the Earth. Ok, so one option that you've got, if you wanted to effectively peel the surface of the earth to make a map, is to cut up all sorts of slits like this, unfurl them kind of like this, and then squish it as little as possible. So I am still - you will end up with these kind of deformations, the skin deforming as you squish it down, especially at the ends, they're not really gonna match together very well. But you've kind of got a sort of flat shape that came from the surface of your sphere. And that really is the idea behind Goode homolosine projection, which sometimes gets nicknamed the "orange peel projection" and you can kind of see why. It does a good job, it's certainly flat, but you have to squish quite a lot of this to get it into the right place. So this map here doesn't preserve direction very well. And also it's not going to be particularly useful for navigating. I mean you're kind of coming off here, and there's this massive gap. It's just not very nice, not particularly nice. So this is a- very squished at the top. So if you kind of look at the top here, the shape of the, of the land, of the coast doesn't quite match the shape that you get up here. But also actually if you sort of imagine folding this round into three dimensions and stitching it back together you wouldn't get a spherical shape. So you can kind of see how things have been literally sort of squished down effectively. So this isn't gonna work very well as a map of the world. So there are a few different options here but none of them are gonna get you completely around this problem of transforming the surface of a sphere into a flat two-dimensional space. So one option is you could say, okay, well around the equator these squares of latitude and longitude are quite square, and that's quite nice. Maybe we want to try and keep that. So when you go up the top, maybe you say, okay, well, why don't we deform these areas so that they're always nice neat rectangles by the time it gets to a piece of paper. For that you end up with something that looks like the equirectangular projection here, which is certainly the world and it's, you know, successfully on two dimensions. But the problem with this one is that direction goes all over the place here. So if you take one of the ones near the North Pole, this line here is actually really tiny when it is on the globe. So if you want to get from one corner of this to the other you'd end up going in like this very weird path, it's not very good for navigation. So that leads to the map that probably most of us know, which is made by instead taking a big sheet of paper and forming a cylinder around your globe. Now you can imagine that you've got a light on the inside of this globe and- that's right in the centre, and shining outwards. And then if you traced the shadow of the land on the paper as it was wrapped in a cylinder around you would end up with the Mercator map projection. This is what Google Maps uses, and it does a really good job, especially keeping the shape of the countries quite nicely, and also in preserving direction. So this is really good for navigation. If you want to get from London to Brazil, you can just head in a straight line. It might not be the shortest way to get there, but you know you're gonna get there. So that's kind of why the Mercator map projection is quite handy. The only problem about Mercator, because you're using a cylinder, things are going to be distorted at the poles much more than they're distorted at the equator. And as a result the area of different land masses ends up being completely, completely mucked up, completely skewed. So if you look on the Mercator map, Greenland and Africa look roughly the same size; but when you look on the globe Africa is absolutely bloody massive and Greenland is teeny-tiny. So it skews your perceptions altogether. Another one is that Australia, sitting just below the equator, the top it is anyway, is actually the same size of the entirety of bloody Europe on the real globe, but I think it just ends up you get a bit distorted when you look at things on a Mercator map. There's still a problem here, right? Mercator's not perfect, there's always going to be a problem. And I spotted something, it's an, it's an, a paper that I've liked for a long time. It's this paper here, it's called 'Orange Peels and Fresnel Integrals'. And essentially in this paper they show that if you peel an orange in a spiral, and then lay it flat on a table, two very nice things happen. First off, as your number of loops in your spiral tends to infinity, the shape that it makes on the table tends to a Euler spiral. Which is this very, very beautiful neat mathematical shape where the curvature changes linearly as you move along it. Really, it's just a lovely idea. Also the amount that your orange peel has to deform to lay flat on the table tends to 0 as your number of loops in your spiral tends to infinity. So I think we've missed a trick here, because I think that this is what maps should look like. I think that we should have, we should prioritise mathematical beauty over geographical practicality. (Brady: What is it doing that other maps aren't doing?) Well it's not gonna be deformed. All of the other maps are deformed in some way. And also it's a Euler spiral, Brady, what more do you want? (Brady: So this is the Euler spiral map? A new way of map. Can we do this?) I don't see why not, I reckon we can cut this one up and see what happens. Like, like the way you would peel an orange, right? Like if you were doing a spiral in an orange, so start at the top and then just wind your way down, I'm just gonna do it on the earth. Start at the top, and work my way down, why not? (Brady: Whatever you do, don't cut yourself.)
- Ok. Did you do a risk assessment Brady? Okay, I'm gonna hit... (Brady: So what's your plan? Are you gonna hit the seam at each line of latitude?)
- And hit this one, Yeah, hit the seam through Russia, ok? (Brady: So ideally you'd be doing this really really fine but you're-) Yeah, I mean we've only got - we've only got this room booked for another hour so- I hope this works. Turns out the world's really big. (Brady: Pretty cool seeing the scissors going through these places) (You're not cutting through Adelaide!) We'll see Brady, we'll see. ...It's literally paradise on Earth.
- (Brady: It ruins everywhere else though.)
- It completely does, it's unbelievable (Brady: ...No, we didn't go there) I don't know if I'm gonna; I think I need to just go a bit further; I'll just go-
- (Brady: No!) I missed it. Technically I missed it.
- (Yeah) Basically... (Brady: And do what you want to New Zealand) You know, I think I'm gonna preserve it. I mean that literally took an hour. We should both have better things to do at our time, Brady. Oh my god, there's so much more of it. I don't think we've got a room big enough. Okay, so you have to remember I have done a finite number of spirals. And the real Euler map projection, as it's now gonna be called, would have an infinite number. What have we done? It's worth it though, Brady, for the mathematical beauty that will unfold. Okay I'm gonna do the other spiral now I think. When I went to university, to study maths, this is not how I imagined my life would turn out. (Brady: You're living the dream) Alright, does that circle look roughly the same size as the other one? Almost, almost... Actually this kind of works, this kind of works. As a map I think people could get used to this. Don't you Brady? (Brady: Have them on the bridge of your ship?) It does mean, you know, if you're in sort of Paris and you want to go to Vienna you're fine. But if you want to go North to, North to London, you do have to do an entire lap of the globe, but you know simple sacrifice. So if you had n loops on your globe, so that the width of one of these spirals is like 1 over n, this deformation reduces like 1 over n squared. So, you know, you're gaining - the more loops you have the more you gain. I think as a map projection, you know, this is mathematically beautiful if geographically impractical. Good solution. Check out these problems and questions. All of them involve the Earth, maps, globes; they're questions that'll really help you get your head around the stuff in today's video and all of them are on Brilliant.org who are the supporter of today's video. Now Brilliant's full of quizzes and courses and puzzles on all sorts of subjects from mathematics and science, but don't feel intimidated, don't think they're trying to test you or make you feel dumb, this is all about changing the way you think, making you smarter and better at attacking problems. But if you don't know the answer, that's fine! There's no judgement, in fact you can go and read a whole bunch of people talking about how they would have attacked the problem. Real mind broadening stuff. Now go to brilliant.org/numberphile to check out all the stuff, there's loads of stuff on the site for free, but if you want to sign up for the Premium Membership, you'll get 20% off if you use /numberphile Our thanks to Brilliant for, well, trying to make the world a smarter place. Hey and speaking of planet Earth, why don't you check out Hannah's latest book, it's called "Hello World". I'll include a link down in the show notes along with a Brilliant link. Hannah: Actually did work out alright you know?
- (Brady: It's fun, it's fun to look at)
Anyone able to code this as a projection for earth? I'm wondering if the land masses from loop to loop would be much closer than we see in the video if you programme something like 10000 loops. Also just what it would generally look like.
"They say mapping the Earth on a 2D surface is like flattening an orange peel, which seems easy enough to you."
Also astute viewers might remember Gaussian curvature from The Remarkable Way We Wat Pizza
Oh man her mispronunciation of Euler is maddening. She studied fluid dynamics, she should know better.