Instant Genius - How triangles are hiding everywhere

Episode Date: June 30, 2024

From tortilla chips and ham sandwiches to teepees and the Great Pyramid of Giza, the world is filled with triangles. But why is this seemingly simple shape so ubiquitous and how do we take advantage of its unique properties? In this episode we catch up with stand-up comedian, mathematician and best-selling author Matt Parker to talk about his latest book Love Triangle: The Life-Changing Magic of Trigonometry. He tells us how triangles can be used to erect the world’s tallest buildings, help spacecraft land on distant planets and create realistic CGI images, and explains how we should all fall back in love with the trigonometry we learned in high school. Learn more about your ad choices. Visit podcastchoices.com/adchoices

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a bite-sized master class in podcast form. Twice weekly you'll hear world-leading scientists and experts talking about the most fascinating ideas in science and technology. day. I'm Jason Goodyear, commissioning editor at BBC Science Focus. From tortilla chips and ham sandwiches to tepees in the Great Pyramid of Giza, the world is filled with triangles. But why is this seemingly simple shape so ubiquitous, and how do we take advantage of its unique properties? In this episode, we catch up with

stand-up comedian, mathematician and best-selling author Matt Parker, to talk about his latest book, Love Triangle, the life-changing magic of trigonometry. He tells us how triangles can be used to erect the world's tallest buildings, help spacecraft land on distant planets, and create realistic CGI images. He also explains how we should all fall back in love with the trigonometry we learned in high school. Welcome to the podcast. Thank you very much for joining us. Oh, thank you so much for having me, Jason. You're more than welcome.

So you've written a book, all about triangles. Are you bonkers? Yeah, it's a good point. And it sounds a lot like what my publisher said originally when I said I wanted to write a book about trigonometry. And they were like, okay, but that's like a famously boring topic. And I'm like, no, that's the point. I want to try and write a book about something everyone thinks is boring, but I think it's super interesting to see if I can do some PR for trigonometry. So we'll all know about Pythagoras theorem at school.

say I'm old enough to have had the old trick charts. Oh, yep, yeah, the tables. Yeah. So was he actually the first to figure it out? And what did he do with that knowledge? It's a good point because Pythagoras is kind of the poster child. They're the mascot. I know they're a human, but they're, let's call them a mascot.

They're kind of like a Colonel Sanders. They were a real person, but now they're just like, you know, a mascot. And they were arguably one of the earliest people to do kind of rigorous mathematical proofs. So instead of just saying, if you have a right angle triangle and you measure the two short edges, either side of the 90 degree angle, and you square those and add them together, you get the square of the long side. And instead of like finishing the sentence there, or then saying in all the triangles we've checked, or this always works, Pythagoras was the person. and it was like, well, can we prove that? Can we show that that is always true? Which is like

half of the beauty of a mass proof is proving it's always true. And the other half is providing some insight into why it's true. And this proof doesn't necessarily give you much of a why, but it definitely gives you the confidence to know it's always true. And prior to that, people had just assumed, it had worked honestly by the time Pythagoras was around, which is well over 2,000 years ago, it had already been working for millennia. So the reliability of what we now call the Pythagorean theorem was not of concern. It wasn't like Pythagoras, you've got to prove this. We think it's going to stop working any day now. But it was that kind of, like the joy of knowing it will always work, which for me is, you know, a lot of the

fun of maths. So let's stick with the Greeks and go on to our comedies. So, funny enough, I was recently in Syracuse. Oh, amazing. Beautiful place of a lot of anyone wants to go. Archimedes is one of my absolute favourites. So what do we know about him in his work on geometry? We know a reasonable amount of what Archimedes was doing, but not as much as I would like. So a lot of his writings have survived, but not all of them. Some have been lost, which is a bit of a shame because he did some amazing work on calculating pie. He was like a pie pioneer, a pioneer, if you will, in terms of calculating pie in, again, a rigorous and systematic manner, other than just measuring it, I'm particularly obsessed with his work at discovering

shapes. So a lot of people will have heard of the platonic solids. That was Plato, who was a bit earlier. Plato decided that they wanted very regular shapes. They want shapes where all the faces are the same, all the edges are the same, all the corners are the same. They're like regular 2D shapes that all have the same edges joined together with these very regular shapes. And there's five of them. Archimedes thought, well, hang on, what if we relax our conditions a little bit? And all the faces don't have to be the same. So instead of having squares in cubes and triangles in like a tetrahedron or like cosahedron, what if they could team up and make like a snub cube or something?

And Archimedes found 13 shapes where you can join regular faces together, but they're not all the same. There might be equilateral triangles with regular pentagons and so on. The only issue is we don't have his original writing. We know he settled on 13 of them because later writing refers to the 13 Archimedean solids or shapes, polyhedron, but they don't specify exactly what they were or how Archimedes decided on 13, which I find infuriating because it's not quite as clear cut as you think. There's some candidate shapes that people argue over like, oh, this one should count

because if you compare like individual vertices, the 3D corners, they're all the same individually, but globally they have different orientations of their neighbours. And some people argue that counts as not regular enough. And some people say that there is regular enough to count. And we don't know what Archimedes' thoughts were. the most recent lost work by Archimedes was found in the early 1900s from memory, where it had been incorporated into another book and people realized the page had previously had Archimedes writing on it, or like a copy of his writing, and we recovered the text

of an original Archimedean work. So I hold the smallest epsilon, if you will, of hope that One day we will uncover his original writings about his shapes, but it sadly feels quite unlikely. Let's hope so. And let's continue a little bit down ancient Greek nerd lane for a moment, if you'll indulge me. It's a big lane. I enjoy it. Where does Euclid come in? Oh, so Euclid was kind of the person who pulled a lot of things together.

So just like Pythagoras was the step change from this seems to always be true. into, oh, this is definitely true and I can prove why. Euclid was just taking that and running with it. It's like, what's the minimum number of assumptions we can start with, and how far can we go proving one thing after another? We prove this, and we can use that fact to prove this, and we can use that to prove the next thing. And so they kind of went through and using the notation in style at the time,

because this is before algebra, and a lot of the things we would normally associate with proofs, and even geometric proofs, they went through and they proved there are only five platonic solids. So Plato didn't miss any. There are only five, and these five definitely work. And there's no arguments. So unlike the Archimedean solids where we're still arguing,

thanks to Plato and then Euclid, we're like, no, it's definitely these five, and we've proven it, which is lovely. So what are those five? So the five platonic solids, some are very common. So they've even got like nicknames, like, cube. We just, we call it a cube. I mean, technically it's got six faces, so we could call it like a hexahedron, but we just say cube, whereas the other ones have slightly more formal names. It's like the tetrahedron, tetra for four, but it's made of triangles, which people, I find that very

confusing. Like, it has got the least triangle name, but it's four triangles. Stuck together. It's a pointy pyramid. It's a triangle-based pyramid, and we call it the tetrahedron, and it's the smallest, nicest platonic solid. Then after that, you've got kind of on its own the dodecahedron. Now, degehedron just means a 3D shape with 12 faces. But this is like the iconic dodecahedron. So it gets called just dodecahedron. Doesn't need any qualifiers. Technically because it's made of pentagons, we should really call it the pentagonal dodecahedron. And because all the pentagons are regular pentagin, they've all gone. got identical angles in every corner, they've got the same length sides. We should really call it

the regular pentagonal dodecahedron to distinguish it from things like the rhombic dodecahedron. Perfectly good. Dodecahedron? Not a platonic solid. And then the last two are just when you combine lots of triangles together. You've got the octahedron. That's, as you can imagine, eight triangles put together. And the icosahedron, which is 20. 20 triangles joined together. it's the Platonic Solid with the most faces, and that's it. You can't make a bigger, and by bigger I mean more faces, 3D shape that's perfectly regular than the Icosahedron. We know that, because Plato went on about it, and then Euclid proved it.

So, just to put this in context, anyone listening who's played Dungeons and Dragons will know these shapes, right? It's the D20, yes. And interesting thing about when you're trying to find dice, the conditions for a dice to work are a bit more relaxed than the conditions for a platonic solid. So the platonic solids are definitely great dice. You'll have your D4 is a tetrahedron, your classic D6, that's a cube. We're very familiar with these because you look at them and they're like, they're so symmetric,

they're going to be fair. You're going to roll this, any faces equally likely. But actually, you don't need it to be that regular. The one property you need is it needs to be something called face transitive, which just means all the faces are symmetrically equivalent. There's like a terrible way to phrase that. If you were to have a shape, you know, it's face transitive, and you're like, that's my favorite face. And then I took the shape away, turned around and gave it back, you'd never be able to find that face again. There's no distinguishing way to determine one face from the other.

and that makes it a very fair dice. So that's why I'm saying there are only five botanic solids. There are more than five perfectly fair shapes for dice. So let's move on from that to more real-world applications. I'm thinking about architecture. So what role does our knowledge of trigonometry play in our ability to build structures? That's a good question. And a lot of people would argue that the dice, the D&D thing,

is the most applied use of shapes and geometry in everyday life. But you're correct. We also like to live in houses and go to buildings and all that kind of stuff. And I find it interesting. You can almost chart the level of mathematics and trigonometry and geometry that a society or a civilization has discovered by how buildings are built. Because the more mass you know, the more certain you are that something's not going to fall over. So before we knew a lot of trigonometry and geometry, buildings were made of massive, big bits of stone, real, chunky, we know it's not going to fall over, but it's what we would call over-engineered. It's just like keep throwing rocks at it until we know it's safe. Whereas now, we can do the calculations. And this may not be the most

obvious bit of trigonometry in structural engineering, but there's a thing called finite element analysis, which is where you look at a whole structure of building. Mechanical engineering, they do this as well. And you want to calculate all the forces. I mean, obviously, you can't do it atom by atom. We do not have the computing power for that. But also, you don't want to do it like whole sections of the building at once, because there's nothing to say, what if that section flexes or changes or moves or twists? You can't assume a giant section is going to be rigid to be considered one bit. So you basically assume the building is built out of effectively Lego where you split the building into a 3D mesh, we would say, in mathematics.

So you're just dividing it up into little chunks, and the chunks are small enough that you can take it to be one solid thing that's not going to change. You don't have to calculate within it, but big enough, that there's enough of them, but big enough that there's not so many, we can't do the calculations. And for a lot of fluid buildings, things with curves, it would be a tetrahedral mesh or some kind of triangle-based 3D mesh for buildings which are much more. angular, then they'll use a cube or a quad or something, a mesh that's better aligned with those edges, which I find particularly fascinating because it's a very similar thing to what happens

in visual effects. So, CGI and computer effects and films, and also everything's approximated as a mesh. It's a collection of very small shapes, which if you're playing a video game, almost always triangles because they're quicker to calculate. So anything moving in a video game, assuming it's a 3D thing being rendered on the fly will be a triangle mesh under the hood. And if you're looking at a film, it's debatable. Technically, they're quad meshes. They're all rectangles. But they're not plainer.

They're not flat. Because a rectangle, the four points in space don't necessarily form a flat surface. They actually do calculate them as pairs of triangles. So for someone who wrote a whole book called Love Triangle, how much I love triangles, I was talking to my VFX friend, Eugenie von Tonzoman, who's done some of phenomenal work. And she's like, oh, we use quad meshes. I'm like, oh, no. She's like, yeah, but they're pairs of triangles. I'm like, ah, excellent. It stays in the book. So I think both cases it's invisible to our eyes. If it works, like we shouldn't know the building was designed

via this finite element analysis. And we're watching a film or playing a video game. We shouldn't be like, oh, wow, this is made of triangle mesh. But without those, you know, invisible seas of triangles, you know, our buildings and our movies and our video games wouldn't be possible. When you need to build up your team to handle the growing chaos at work, use Indeed-sponsored jobs. It gives your job post the boost it needs to be seen and helps reach people with the right skills, certifications, and more. Spend less time searching and more time actually interviewing candidates who check all your boxes.

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So this is fascinating. My obsession with navigation is of the very, very slow kind, where you want to move slowly, calculate your own triangles, and make sure you know where everything is. Now, obviously, the end result of that is things that people can use for much faster navigation. But kind of the history of knowing where we are is the history of, you know, getting a lot of triangles. So the meter was defined as, well, it's one 40 millionth of the circumference of the earth. But to do that, we had to know how big the earth was, which meant we had to know where a lot of things were very, very precisely.

And so there was a chain of, this is just over 200 years ago, a chain of over 100 triangles that went from Dunkirk in the north of Paris up in France, all the way down, directly south, finishing around Barcelona. and there were so many triangles because it's kind of easy to measure angles. This is the great thing about triangles. Angles, easy. Doesn't matter how big your triangle is, the angles don't change. You can measure the angles in a triangle very easily. You use a thing called a repeating circle, this is what they used, which is where you kind of measure the same angle over and over and over again,

and the machine mechanically accumulates the running total of degrees. So you measure, I don't know, say 10 times, and then you divide the accumulated angle by 10, and you get a nice accurate measurement. The difficult bit is measuring a distance. And to do that, they had to get down on their literal knees and measure the length of a road by moving platinum rods. They had to get rulers. There's one very straight road just south of Paris.

There's another one in the south of France. And they measured these rules very, very carefully. And then they were able to calculate those distances, and then you can triangulate your way back to everything else. and you can still see the scars, the monuments to this in the UK. There was a thing called the Great Triangulation, great name, started in the 1930s. There have been previous triangulations, but this was the Great Triangulation of Great Britain. And if anyone in the UK is familiar with their OS maps, the modern OS map was thanks to the

Great Triangulation of Great Britain. And to this day, there are still hundreds of trig points. These were these altar-looking stone monolithic things. Normally somewhere scenic because they had to be able to see at least two other ones to measure a giant triangle. So they're normally on a hilltop. And now they sit there and the vast majority have no indication what they are. They just look like this ancient altar or something that's been left there. But they are the vertices.

They're the corners from the great triangulation of Great Britain. And there's a hobby called trig pointing, which you can imagine. I'm a big fan, trickpointing. UK, if you want to get involved. And people see how many of these they can visit and you take a photo of you with the trigpoint and you can log it on the website. That's a long way to say,

that's my favourite type of navigating because it enables all of the navigation to be able to locate and visit these trigpoints. So let's move from the Earth into space so we can also use this idea to do all sorts of fascinating things in space. Yeah, it's basically the great transformation. But now it's in space and global.

If you want to find your global position via a system, GPS is the way forward. There's a few complications, though, but like the maths of the triangles is pretty much the same. The slight complication is the satellites are not on the earth and they're moving fast. So before your triangulation stations, your trigpoints, they're not going anywhere. They're not moving and they're at the same distance. from the center of the earth as us, which may sound like I'm being overly pedantic, but they're both very important points, because as we discovered in the early 1900s, thanks to our good friend Einstein, there's both special relativity and general relativity, which indicate that time, time is not

as absolute and as straightforward as we would like. And when we put a satellite in space, and it beams out a very precise time-coded signal, And by having a handheld device that can receive those signals and looking at the slight timing differences between the signals, that gives you your relative position to these fixed location, or rather known location, satellites. But to do that, you need to have a very reliable, precise, accurate, both those things, signal coming from the spacecraft, and you need to know the time on the spacecraft relative to the time where you are, which you think would be the same, but it's not.

not, partly because the spacecraft are moving. So they are moving faster than we are, and we know from special relativity, that if you're moving fast, or you've accelerated relative to someone else, you will experience time at a different rate. So because the spacecraft are moving quite quickly, the time on the spacecraft is passing slower. So you're like, oh, great. So now the clock on the spacecraft thinks time is moving at a slower rate than we do, which is going to mess up our signals. But, thanks to general relativity, we know if you're in a gravitational field, the rate of time changes. And the closer you are to a very large object with a lot of mass, or just an object with a lot of mass, the slower time will pass, famously in the

film Interstellar. And if you like tying a neat bow on things, my friend Eugenie, who I was talking to about the triangle meshes, did the VFX for the black hole in interstellar. Because if you You need a VFX person who knows their maths. Eugenia person. So in that film, as a plot point, when they're on a large planet, time slows down. Now, from the satellite's point of view, we are closer to the Earth. So because we are further into the Earth's gravitational well, our time is traveling slower than the satellite, which is the opposite direction to the speed dilation.

So actually, the speed dilation drags us a couple milliseconds in one direction, but then the gravitational well dilation drags us back in. overshoots in the other direction. So we have to understand both these types of time dilation, which I should point out involves a good knowledge of triangles and trigonometry, to then compensate for the time change, to then do the trigonometry to triangle at our location. So in GPS, it's just a fantastic use of mathematics. There's also some physics and engineering, but it's mainly maths. Okay, so for my shame, I'm a physicist. Well, I'm not. I studied physics. So you talk about something I find really interesting. I'm also a musician.

So I'm really interested in sine waves and Fourier analysis. What can we say about that? How does that create the sound? It's such an interesting area of mathematics. And you're right, it's very physics-y. I will allow that. I married a physicist. I'm surrounded by physicists. And actually, you know, well, I did first come across Fourier analysis when I was doing a physics unit at university. So I do actually have to thank physics for introducing me to Fourier analysis. That was in the context of like laser light. But anything that's a wave, which be it, you know, light or a signal, audio, as you're saying, it definitely links into music. The thing called Fourier analysis basically says you can take any signal and you can decompose it into pure.

were sign waves. So when I'm talking now, and wherever you are listening to me, the sound I'm making is, if you were to look at the sound wave in like some software, it would be jaggedy all over the place and not just because I've been talking too much recently, so my voice is croakier than average. In general, anything that sounds like some kind of texture, it's a real jaggedy, messy soundwave. Whereas a sine wave is a pure wave, like it's perfectly smooth, it's up and down, no variation, pure sign wave. It's what we call technically annoying to listen to. It's just that pure tone, which I will not attempt to recreate because it would be a very rough approximation. Now, the big inside of Fourier analysis is you can take any complicated signal and you can decompose it into a combination of sine waves.

And kind of the big extra step on top of that is, it's all well and good to know that's possible. mathematicians then proved, like, it's definitely possible every time. Like, there's not like some weird way we're going to come across one day where it won't work. And we've discovered techniques to do it because it's all well and good having an existence proof this can be done without being able to actually do it. So now we can do that. And that's got all sorts of useful applications. The original application, Fourier themselves, the reason they came up with it is they were holding a metal rod. It was like a poker that you would use in a fireplace to poke the fire.

they were holding a metal rod, a poker, and they had one end in the fire. It's obviously very hot. They were holding the other end with their hand, so it was warm, but not too hot. And they thought to themselves, I wonder how long it will take for my end of the metal rod to get so hot, I can no longer hold it. So how long does it take the heat to work its way up the rod? And heat in a metal is basically moving, because sound wave, there's waves of movements of the atoms. And so Fourier came out with Fourier analysis as a way to analyze. heat movement in a metal rod. We don't use it that much for metal rods anymore, but we still do a bit.

As you were saying, it mainly comes in for signals and music. So in terms of signals, if you're listening to this, you know, on a smartphone, it got to your phone because we can put data into waves and we can use Fourier analysis to pull those waves apart afterwards and extract that data. And without that, our modern communications wouldn't work. Music-wise, I mean, the crude example, example is the little equalizer, the visualization on an old school stereo, and little bits going up for different frequencies, that's for your analysis. It's like showing you how much of each frequency. I mean, that's a, you know, very big categories because you want how much bass, treble, so on. But that on a very fine level is what we're doing. And I find it, you know,

absolutely amazing that we can take, no matter how rich and wonderful the sound of something is, and its heart is just sine waves. And understanding how those waves combine allow us to do also, sorts of analysis and processing. I mean, if it wasn't for Fourier analysis, we wouldn't have been blessed with autotune. So, you know, we've got a lot to thank it for. So we've covered an awful lot of ground there. Essentially, what we've done, we've championed maths. So I'm thinking about especially younger listeners, what can you say to encourage them by way of summary of studying maths, you know, putting in the effort? Well, it's similar to what we've just done. And we've kind of recreated the book. Because when I went to my publishers and when I want to do a book about

triangles. I had to then sell them on the journey. I'm like, I want to look at triangles and geometry and calculating distances and angles, which are all kind of easy to get your head around, the sort of stuff you learn, primary school into lower secondary school. And I'm like, but once we have triangles, then we get trigonometry. And I want to do all the trigonometry stuff, which is like later in high school, you're looking at trigonometry and all these other uses. And as you and I had our sign tables and to look up values at school now, obviously calculators are doing a lot of the heavy lifting there. And then later on, when you go to university, it's four-year analysis because now we've got sign waves.

We've got sign functions. We've got trigonometric functions because of our knowledge of triangles. The wonderful thing about maths is seemingly unrelated things often have the same logic behind them. And what we learn from triangles, it turns out, describes waves. And waves could be argued triangles, like behind the scenes. And so I was like, then in the book, I want to talk about waves and Fourier analysis and data signals and all this. But if you start there, like, if I'd pitch that, they're like, it's a book about triangles, why are you talking about music? And I'm like, no, no, no, it's a journey. And so that's kind of my moral for both my publisher and for young people studying mathematics. It's a journey. And that cuts both ways. Sometimes you're learning maths at school and you're like, this is boring. Why do I have to learn this? But you need all the bits before you can move on to the next bit. It's such a progression. And so sometimes you've just got to master and learn.

bits of mathematics because you're going to need it in the future. But then the other side is, there's always some kind of fun use and fun application for every bit of mathematics when you're doing it. It's a bit like playing sport. Yes, it's boring when you have to do the drills and the exercises to learn it. You can't skip that. But once you've got the tools, it's worth remembering as a lot of fun things you can do with it. They're the two things I've tried to get across in this book. On one hand, no matter what level the maths is, there are some fun and interesting and unexpected applications or uses or things you can do with it. And at the same time, it's leading to more maths. We're constantly learning new maths, discovering new shapes,

working our way up that hierarchy, which I find endlessly fascinating. And I think it's a shame if students are at school and they don't see either of those. They don't see what they can do with the maths they're learning now and they don't see what they're going to be able to do with it to learn new maths in the future. Thank you for listening to this episode of Instant Genius. the team behind BBC Science Focus. That was stand-up comedian and mathematician Matt Parker. To read more about the topics we've just discussed,

check out his latest book, Love Triangle, The Life-Changing Magic of Trigonometry. If you liked what you just heard, please do consider subscribing to Instant Genius on your preferred podcast platform. The current issue of BBC Science Focus magazine is out now. Pick up a copy wherever you buy your favourite magazines or download us on your app store of choice.

You can also find us online at sciencefocus.com. This podcast is sponsored by Name, Audio and Focal. The texture and emotional depth of music can be lost through digital sources or poor signal. Name Audio believes you can have digital precision with analog warmth. Alongside French acoustic specialist focal,

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