- A roulette is a curve obtained by rolling one curve along another. This simple idea leads to some lovely curves, and I'm going to tell you about some of those today. So we'll see how to construct this beautiful design. We'll meet the curve so appealing. It was given the nickname, the Helen of Geometry, and we'll see examples of a roulette curve that's so useful, it has applications in everything from gears to nuclear power plants. So let's start then with the simplest kind of curve rolling along the curve. So what's the simplest curve we can imagine? Perhaps a circle, right? Simple. There's even a simpler one. If you think what actually is a curve, it's a path made by a point moving in space somehow. So the very simplest thing, a point moving in space could do is to go in a straight line. So the first curve we're going to look at, first of these roulette curves is the cycloid. And this is made by a circle rolling along a straight line. So we think of a wheel rolling along a road, and we are following the path of just a point on the rim of the wheel on the circumference of the circle rolling along. And we can see that as it rolls, we get these arch like curves being produced. And that's the cycloid. Now wheels have been around for a long time for thousands of years. It's perhaps surprising that the first record we have, the first evidence we have for someone thinking about this curve is only in 1501 when Gilles de Roberval was working on various mathematical problems, including he was trying to find a way to square the circle, which he didn't know at the time. We do now know is impossible. There's no way to do that. And so, he was trying various things. And this was this curve, this idea of rolling a circle along a straight line came up in that. He didn't manage to square the circle. He also didn't get too far with understanding this curve, the cycloid, because he thought that those arches, he thought those might be bits of a circle. And I mean, we can see in this picture that they're not. Now, we are lucky 'cause we can just program a computer to draw these things. But for him, it's a reasonable guess. Later on, the mathematician, Marin Mersenne, and he thought that perhaps the arches of a cycloid were half ellipses. Now again, this is a pretty good guess, and it's better than thinking they're parts of circles. And if you look at the diagram there, I've own half an ellipse in red and the cycloid arch in blue, and you can see, they really are very close. So I think this is not a bad guess from Mersenne. So when you've got the curve and you're interested in it, what might you ask about that curve? Well, let's think about circles. We know, we all learn at school the formula for the circumference of a circle. If the radius is R, the circumference is two two pie R, and we know the area of a circle, pie R squared. So we know those formulae. And so you might ask about cycloids. So I mean a cycloid goes on forever. So if the whole thing is infinitely long, but if we just look at one of the arches, we could ask, how long is that arch? How long is the length there? And perhaps what's the area underneath the arch? Now, Galileo was thinking about these questions and it's actually Galileo who gave this curve its name, the cycloid. He found a very practical way to try and estimate the area under a cycloid arch. He actually got a bit of metal, he made a circle, then he rolled it along another bit of the same metal, the same thickness, and he marked out a cycloid and then he cut up them out and he weighed them, right? Very practical. This is a applied maths or something, right? So he weighed them. And of course, if you've got two bits of the same material, the same thickness, then the weights of those, what they weigh will be proportional to their areas. And he found that the area of that cycloid arch, the area underneath it was about a subject, bit of an accuracy in the cutting outs and everything. It seemed to be roughly three-ish times the area of the circle that made it. Now, if you are trying to guess making a conjecture, some number around three-ish that's got something to do with circles then something pops into your mind. So he thought maybe it could be pie times, but he couldn't quite resolve it. So that was a puzzle. And we want to know what this is, and we're interested in perhaps the length of this thing. So what's the area under this cycloid arch? Galileo tried it by experiment. And he told Mersenne about this. So Mersenne hears about cycloids from Galileo, and Mersenne is a very important figure in Europe at the time in terms of mathematics, because he's sometimes called the post-box of Europe. He corresponded with dozens and dozens of people. Lots of mathematicians all around Europe were in correspond with, or if they happen to be living in France, actually meeting up with Mersenne and telling him about what they were doing. And he would say, "Oh, did you know?" So he told Gilles de Roberval,"Hey, Galileo has been working on this curve, this thing called a cycloid, and he's been working out wondering about the area under the arches. And so, Mersenne would pass on all this kind of mathematic gossip, I guess, to various people all over Europe. And this is a really useful thing. It's precursor, I suppose, of what we have perhaps now with academic journals, where you say, "Look, I'm doing this." And then everyone else has a look, and then they might try and extend those ideas. So Mersenne is very useful. He's also, I have to say, he's a good mathematician in his own, right? So you may well know the name, Mersenne from at least two contexts. So if you happen to see any of my lectures on mathematics in music last year, you'll have heard of Mersenne laws, which tells you how the frequency of the sound made by a vibrating string varies with its length, its tension and its linear density. Yes, got that right. So Mersenne laws tell you something about maths in music and he investigated that area. We also may have heard of Mersenne primes. So Mersenne studied prime numbers. He was interested actually in perfect numbers. And he studied primes that are of the form, power of two minus one. So seven, for example, two cubed minus one. So we perhaps have heard of Mersenne prime, so he did a lot of his own mathematics, but he also had this very important role of passing along information between people. So Gilles de Roberval, he actually did find a formula for the area under a cycloid arch. And it's a really lovely thing, perhaps slightly unexpected, especially since I led you down the garden path. The area under a cycloid arch is exactly three times, not pie two, three times the area of the circle that generates it. To me, surprisingly, it's just a whole number multiple of the circle area. So Gilles de Roberval found that out. Now, he was a curious character, Roberval. He held his information quite jealously because he wouldn't actually reveal, he wouldn't often even tell people what he'd managed to prove. And if he did tell them he'd managed to prove something, he wouldn't necessarily share the proof with them, and this was a rather idiosyncratic thing, of course, but there was method in the madness. Roberval's job, he had a professorship, and this professorship was renewable every three years. And the way they decided who would be the next professor for the next three years was there was a competition, an open competition with some mathematical problems, and you could submit your answers to those problems, and if your solutions were the best, you got to be that professor for the next three years. So the twist is that the problems in that competition were set by the incumbent professor. So there is a great incentive if you have a particular thing that you know how to do. You don't actually want to tell anyone else how to do it, because then you can set that problem every three years, and you are the only one who knows how to do it, so you win the competition that you have designed, and you are entering every single time. So there is a bit of an incentive for Roberval to have some secret techniques, but even so, he was especially secretive about his work and a lot of what he did wasn't published until a long time later after his death. But we do know that he was very likely the first to find the area under the cycloid arch. Now, it wasn't just Roberval, and Mersenne, and Galileo. Anyone who is anyone, think of a mathematician you've heard of from this kind of time, and they almost certainly worked on cycloid at some point. They are so appealing to people. The statue there, you can see, that's from the Louvre, that's of Pascal. We may have heard of Pascal's trying on the things he more or less invented with the Fermat, the science of probability theory, but he also, he worked on the cycloid, and it happened like this. He decided actually at one point,"I'm not going to do maths anymore. I'm going to focus on theology. I think that's what I should be doing." And then one night he had a terrible toothache and he couldn't sleep. So he did what sensible person would do. He thought about lovely mathematics to distract him. And the lovely mathematics he thought about was the cycloid. And actually, this statue of him, depending on where you are sitting, you maybe just be able to see that the stone tablet he's looking at has a cycloid on it. So this is the Descartes, so pleased when I us to show you. This is him studying the cycloid. That night, what he was thinking about, the cycloid, it was so distracting that the pain went away. And so, he took this as a sin that God was okay with him doing mathematics, and we're glad he did because he did some lovely mathematics after that. He went on to think about cycloids for eight more days, and he proved a lot of things during that time. So this is a very distracting and appealing curve. That's one reason why it's called the Helen of Geometry, Helen of Troy, the face that launched a thousand ships, famous beauty, but not just for being beautiful, for causing, let's say, squabbles. And this is another reason why this cycloid has got this nickname because so many people studied it, and then they argued about who done what first and whose ideas were better. So Descartes and Fermat both studied the cycloid and they were trying to understand properties of tangents to the cycloid. So the tangent to a curve at any point is a line that just touch it. It doesn't cross it, it just touches it. They both had studied these things. They didn't like each other's methods. Descartes said that what Fermat had done was ridiculous gibberish, which isn't absolutely the polite way to speak of a fellow mathematician. Descartes didn't like what Fermat had done. Roberval, here he is again. Now, remember, he didn't tell anyone what he'd done or he didn't like to explain what he'd done. So when the Italian mathematician, Torrichelli, independently came up with a way to find the area under the cycloid arch, Roberval got really crossed. He went around saying,"Torrichelli's stolen this from me. He must have stolen it somehow." We don't know, Roberval didn't tell anyone what he'd done. And then the rumor went round that Torrichelli was so ashamed when he'd been caught plagiarizing in this way. Don't think he did, but he was so ashamed that he'd actually died of shame. Now you can judge for yourself whether this is true or not. I do have to say, at the time when he died of shame, he also happened to have typhoid, but we know, let's not make any assumptions. So all these arguments and many others were how this beautiful curve got to get it's nickname as the Helen of Geometry. But the story of the cycloid doesn't end there. I want to tell you something about clocks. So in 1656, there was a breakthrough in clock design. Before 1656, the best, most accurate time keeping device was a sundial. Not very useful if you are indoors or if it's nighttime, or if you live in Britain, where it's often cloudy, but sundials were the most accurate things that we had. They were mechanical clocks, but they were, I don't want to be rude, but they were rubbish because they lost up to, or inaccurate up to about 15 minutes a day they could lose. And so, this wasn't good enough, and people were trying to think of mechanical clocks that would be better at time keeping. And 1656 was the year when Christiaan Huygens, the Dutch scientist and mathematician produced and built the very first pendulum clock. So this was a great leap forward. What's good about pendulums? Why would you use a pendulum in a clock? Well, I'm not going to go into like the full details of it, but I want to just show you a bit of what's going on mathematically, because then I want to explain why there might be a way to improve upon it even when you've got a pendulum. So what happens to the pendulum? So you've got a rod or a wire or a string, something like that fixed at the top. And then at the air bottom, it's got a weight. We call it the bob usually, and then it will swing backwards and forward. And if you just let it go and it falls under gravity, the bob will move in archs of circles, and it move backwards and forwards. It'll tick and then it'll tock, come backwards and forwards. And what's great about pendulums is that this movement, backwards and forwards, is very regular, right? So you can literally set your watch by it. Now, why is it regular? Well, again, I won't go into all of the details because to do every line of what I'm about to say would need a bit of calculus. So we won't do that, but I just want to give you an indication. So what's happening in a pendulum? When you've got a pendulum and you're going to hold it here and then you're going to release it and let it move just with gravity. So maybe it's an angle to the vertical. When you set it going, that angle could be called theater. And then there are some forces acting. If there's no forces, there's no movement. So what are the forces? Well, you've got gravity. So force is mass times acceleration. What's the acceleration here? It's the acceleration due to gravity. That's what that little G is. And then the M is just the mass of the bob, whatever it is. So that's the force downwards, vertically downwards. And this thing, it's joined on, the bob is joined to the top of the pendulum. And so it can't get any further away from the center of that circle. So it's got to be something pulling it towards the center of the circle. That's what I've called T there, the tension in that string or rod. So that's pulling you towards the center of the circle and exactly counteracting the bit of the downward force that's trying to move you away. So what is left? So those balanced out. Then the only force that they're sort of left over is the force that kind of pulling you along the arch of the circle. And if you use a bit of trigonometry, you find that actually that's related to the sin of theater, the sin of that angle. So at this point, and if you've ever did study earlier on pendulums, you will have made an approximation. And it's this, when theater is very small, sin of theater is very close to being theater. So you approximate, you make an approximation. And the reason you do that is because you can't solve the equations if it has sin theater, but you can't do it if it has theater, okay? So then you do a bit of equation solving, a bit of algebra, a bit of calculus, and you can get an expression for the period of the pendulum. What's the period? It's the amount of time it takes to get from your starting point over there and back again. That's the period. Back again. And when you do that with this little approximation, this is the expression you get for the period. Doesn't matter exactly what it is for these purposes. But what I'm want you to notice is, so it's got a pie in it. It's got an L, that's the length of the string. I didn't say that. And it's got gravity. This does not depend on the angle that you started with. It doesn't depend. So what this tells you is, whatever height you are, whatever angle you're at, when you release the pendulum, it takes a same amount of time to get back. And that means if you set it going, then over time, the system loses a bit of energy. Maybe it's moving to a smaller angle, but that doesn't affect the period. So it still keeps ticking and tocking at the same rate whatever the angle is. That's great, except for this fact that I mentioned, all of this depends on an approximation. So it only is truly accurate for small angles. And as soon as the angle gets a bit bigger, it becomes less accurate. So these pendulum clocks were great, fantastic innovation. They now meant that you could produce clocks though accurate to 15 seconds a day, not 15 minutes a day, right? This is a huge improvement in clock design. Huygens wasn't satisfied with this. I mean, it's great, but he thought, he was aware of this approximation that had to be made. He thought, what would be great is, this pendulum is moving along the arch of a circle, and approximately has a constant period. It doesn't matter where you start it going from, it reaches the bottom in the same time, right? If only there was some sort of curve that I could get my pendulum to move along, which really did have that property, that it really did always reach the bottom in the same amount of time. And if you take the Greek for the same amount of time, that's tautochrone. And so this thing is known as the tautochrone problem. Find a curve, such that wherever a particle is release on that curve and it's just moving under gravity, it will reach the bottom in the same time, and then sort of bisymmetry, it'll go back up to the opposite point and then come back again. So with such a curve, if it exists, if you could get a pendulum to move along that path, then you really would have a constant period, not just an approximation to one. So Huygens saw, wouldn't it be lovely if there was a curve like that? Can you guess what curve that might be? I mean, you can see it here. It's actually a cycloid. I mean, it's what you know. I've turned it upside down, but it's exactly the cycloid that we've seen, which is like astonishing the first time you see this, because the cycloid just is like a circle rolling along a straight. Why that be the answer to the tautochrone problem? That's one of the wonderful things about mathematics that you sometimes have these beautiful coincidences. So this is the solution to the problem, but we are not done because that's all very nice. How on earth do you get a pendulum to move along the cycloid, right? If you just set it going, it'll make an arch of a circle. How would you get it to do this? So for that, we need to think about the next kind of roulette curve. So roulette curves, remember, are curves that are made by rolling one curve along another. And this kind of roulette curve is called an involute, an involutes of what you get when you roll a straight line along curve. This gift was going a bit faster earlier, but this is an example here. We're rolling a straight line along a circle. So on the graph here, you've got this red straight line, which you can think of as being like a thread that's unwinding gradually from a roll or threat or something like that. And you're following what that straight line does, a point on the end of the straight line, a line segment, I guess. You're just following its path as it moves around. So these lines that are rolling along the curves, they're going to be tangents to the curve, I guess. So you can either think of it as that rolling a line along, or you can think of it as unwinding a taut thread from the curve. So this is the involute of a circle that I'm showing you here, looks a bit like a spiral. So what's this got to do with the pendulum problem? Well, there's an involute of circle just there. So you'd have to gradually watch it going round. So we've produced these curves. Now, if you imagine, just turn that around a little bit. So here's a circle. If I had just a circular thing in my clock somewhere, and I wrapped my pendulum around it, so you fix the pendulum and I guess, you know, it can't just be a rigid rod that's holding, it'll have to be a string or a wire that can bend. So you wrap it around the circle. And then when you release, the bobs at the bottom, when you release that, it'll unwrap itself from the circle as it goes down until I guess in this diagram, until it reaches the vertical, and then it'll start moving in some freeway. But for that first bit of the motion, it's unwrapping itself from the circle, and it will produce exactly this involute. So if you wrap your pendulum around a curve and then release it, you'll get the involute of that curve. So now what we need to ask is, is there a curve, right? X, some mysterious curve whose involute is a cycloid'cause then instead of a circle, we could put that there, wrap the pendulum around it, let it go, and it would form a cycloid path. So guess what? Guess what curve has an involute that's a cycloid. It's a cycloid. Cycloid is down to everything. Yeah. The involute of a cycloid is a cycloid. So what you can do is, you can get two cycloids, put them next to each other, then wrap the pendulum around, and you have to have to make sure the length are right. It has to be half the length of this arch. And then, when you release it, it will make a nice cycloid path. So this is exactly what you need to produce at the perfect pendulum clock. Okay. Slight hitch. With real world, when you actually try and build these things, you find that the amount of extra friction they introduce into the system balances out the improvement that you made in the theoretical design, but, okay, nevermind. It's a lovely bit of mathematics. So these, cycloid are great. The involute of a cycloid is a cycloid. Well, since we are talking about the Torrichelli problem, I want to mention Brachistochrone problem, which depending on how good your Greek is, mine isn't very good. But I happen to know that Brachistochrone means kind of shortest time. And this problem was posed by Johann Bernouilli in 1696. And he said, "Okay, what about, if we have a slightly different situation, we're not interested in the curve that has the same time to get to the bottom wherever you start from. What we're interested in is if you have two points, top on a bottom point and you want a particle that's falling along a wire between them, you want it to get to the bottom in the quickest time. What shape should that Y be? What curve should that be?" So you could think of various guesses like parabola, maybe, I dunno, lips, something like that. And Johann Bernouilli posed this problem, sort of an open problem to mathematicians. What's the answer to this? What curve should I pick? And several people managed to solve this problem quite quickly. So Johann himself solved it. Johann's brother, Jacob solved it. Leibniz, his girlfriend Leibniz, one of the inventors of the calculus solved it. And there was a mysterious anonymous entry to this problem. And when Bernoulli saw the entry and he saw the lovely mathematics that it had used, he said,"I know. I know who did this (speaking in foreign language). We know the lion by his claw. The mathematics are so good, it could only have been one person who did this, Isaac Newton." And it was. So Newton submitted an anonymous entry, and so all of these people solved this problem. Guess what curve is the solution of the Brachistochrone problem. It's a cycloid. Again, it's amazing. So a cycloid is wonderful. We all love the cycloid now. It is also, not only does it reach the same amount of time, but it's the best possible. It's the quickest time that you could possibly have of all the possible poles you could take from one point to the bottom. So the cycloid, there it is. It strikes again. Now I mentioned before that Gilles de Roberval managed to work out the area under a cycloid arch. And I want to show you his argument. And I will say that this was before calculus, it was before a lot of modern techniques. This argument though, it has good bones. It's basically the right idea. It's a really neat little argument. Nowadays we wouldn't say it's totally rigorous, but it can be made rigorous, and that's what I mean by saying, it's got good bones. You can make it rigorous now with calculus, and it still works, but I wanted to show you it'cause I think it's a very lovely argument, really clever. So remember with a cycloid, we've got a circle rolling along. And if you start with a point at the bottom, and you start rolling along, then at that point, at some point it will reach the highest place it could possibly go. And that's after the circle has kind of gone half way round. So in the distance between the point being at the bottom and the point being at the top, the circle has rolled half the circumference along the road. So if the circle has radius R, that point at the top is two R the diameter above the road. And how far has the circle traveled? It's traveled half the circumference. So it's half of two pie R, so it's traveled a distance of pie R, so just remember that for later. So what Roberval did is quite ingenious. He created something. So he took half the cycloid arch, And then of course, if you know the area under that, you can double it and find the total area. So once he got here, he's created this thing called the companion curve. So you take a semicircle, so your original circle, we just take half of that semicircle sitting there at the beginning and a semicircle. It's made up of a whole bunch of lines stacked on top of each other, like any shape. And of course, if you want to know the area of that, you add up all the little lines in some kind of infinitesimal way that we won't go into. But this is the same shape we take. So then what we've got in that picture, you've got the blue line, that's your half cycloid arch, and you're creating this new curve called the companion curve. And what you do is, you take at any point as you're going up, you take the point on the cycloid arch and you shift it to the right by exactly the width that is the width of the semi circle at that point. So you can see all of these green lines are just being shifted along, and that's how far you move to right. And when you do this, you create this companion curve. Now, it's actually a sin curve. We don't think Roberval necessarily knew that, but you can prove it nowadays that that thing you produce turns out to be a sin curve, which is already quite cute. So remember, we're trying to find something about the area. The area off this new shape, we've made this kind of little curvy shape, the bit that's bounded by the cycloid arch and this companion curve. There's something called Cavalieri principle, which says, if you've got two shapes and whenever you take a slice through them, the line you get has exactly the same length in each shape, then areas of those shapes are the same. And that's something you can prove with calculus, but this was in use before that, and it's plausible, but also happily true. So with that, given the way we've constructed these shapes, it's clear that the semicircle has got the same area as this new strange companion curve shape because we have just got exactly, by creation, we've done this, we've made those lines be all the same length. That's how we constructed it. So those two things are the same area. So I've labeled them a for the area of the companion kill bit and S for the area of the semicircle, they are the same. And if the circle has radius R, the total area would be pie R squared. So the area of the semicircle is half pie R squared. So that's one bit of the area we're interested in. We'll just move that over to the side for a moment. Let's think about now, I've enclosed that half cycloid as you see in a rectangle. And here's a rectangle, that same rectangle, I've got the companion curve in the middle of it, and lovely blue and yellow colors'cause they're best colors, and there's the companion curve. And you can sort of plausibly see that bisymmetry, right, we made this thing by rolling a circle from left to right. If you turn it the other way up and roll it from right to left, you get the same curve. And by the symmetry of the situation, that means that this rectangle has a rotational symmetry. That when you turn it around, you get the same picture. And that means that this curve divides the area of the rectangle exactly in half. So the blue area is the same as the yellow area. What is the area of them then? Well, we talked about this rectangle. The base of it is pie R because it's how far the circle rolls when it's gone half the way round. And the height of it is two R, the diameter. And so, the area of the whole rectangle would be base times height, two R times pie R, which is two pie R squared. And we want half of that. So the bits I've labeled B and C have equal area. Each of them is half of that, so each of them is pie R squared. Okay. We're nearly there. All we need now is to add together A and B. So what we want for our half cycloid, we've got the bit under the companion curve, and then we've got the bit between the companion curve and the cycloid. So that's A plus B. A is half pie R squared, B is pie R squared. Add them together. You get one and a half pie R squared, and that's half the answer we want because we want the whole arch. We know half of it is one and a half. So all of it is going to be three pie R squared. Hey, so that's what Roberval did. And I really like that little argument. Many other people worked on cycloids. If I want to talk about any other roulettes, this will be my last slide on cycloids. Many other people worked on them. So you might recognize the chap on the left there with St. Paul's Cathedral behind him, that's sir, Christopher Wren, he worked on cycloids, and he managed to prove that the arch length of a cycloid is exactly another lovely whole number multiple eight times the radius of the circle that creates it. So that's very nice. Cycloids didn't just catch the imagination of mathematicians either. They appear in several very famous works of literature, which we all have either read or think we've read. So "Moby Dick", there's a whole bit about cycloids in "Moby Dick", which is very interesting. They're mentioned in "Tristram Shandy"? They're mentioned in "Gulliver Travels". So obviously, these are things that are in people's minds this time. And so, there's this advert here. Next year, it's going to be the 300th anniversary of Christopher Wren's death. And he was a Gresham professor, not of geometry. He could easily have been that, but he was Gresham professor of astronomy. And so Gresham College are going to be doing some events around Christopher Wren. And so if you're around in 2023 here in London, or even if you want to watch online, there'll be some events around Christopher Wren to mark that 300 years. And the other thing next year is, obviously you are thinking now,"My goodness, these links between mathematics and literature, if only someone would write a book about the wonderful links between mathematics and literature. And if you are thinking that, surely you are, then you will delighted to know that in 2023, my book on mathematics and literature, which is called "Once Upon a Prime", already it's great title, right, is coming. So that's for next year. Okay, cycloids. We've talked about cycloids, they're brilliant. We roll a circle along the line. What else can you roll a circle along? You can roll it along another circle. And when you do that, you've got to decide, am I going to roll it along the outside of the circle? In which case you might get something like this called a cardioid 'cause it's sort of heart shaped, or am I going to roll it along the inside of a circle? And those things. So outside of a circle, epicycloid, inside of a circle, hypocycloid. So you've got some decisions to make then. Do I roll my circle along a circle of the same size? If you do that, you get this thing, cardiod, or shall I make the thing I'm rolling it along bigger? And if you take sort of whole number multiples of the radius, you get some quite good things. So there's your cardiod. Perhaps you could draw your circle along one that has twice the radius. Then you get what's called a nephroid. So that's from the word nephroid from kidney. Well, it looks a bit like a kidney, I suppose. So you'd find that if you do this, with the nephroid, you get two arches with this. Cardiod, you get one arch, and you can, if you take higher and higher multiples of your radius to roll along, you get more and more arches. What about if you roll your circle along the inside of a circle? So I'm not going to talk in detail about all these curves, but if you roll your circle on the inside of a circle, then you start to get things that look a bit like if anyone has a kid, had a toy like a spirograph or various other toys like this, what you do is you you get a little circle and you put your pencil in to a small hole just inside the rim of that little circle, and then you kind of roll it round and round and round the inside and you get these beautiful pictures. I've got some on my top today. And so, these things, well now, they're not quite hypocycloids. The reason is that the point who's part of your tracing is not on the circumference, it's a little inside. So there's a word for when you do that. And it's this word? Well, trocoid is involved. So when we did cycloids, circles rolling along the line, if that involved a point on the circumference, if you take a point not quite on the circumference, a little bit inside, maybe, or sometimes a little bit outside, if you want, what you get isn't a cycloid, but a trocoid. There are various possibilities for what you get based on how far away from the circumference your point is. So by an analogy, you've got epicycloids rolling along the outside of a circle. Again, if you have the point not quite on the circumference of the rolling circle, then it becomes an epitrocoid. And so, what spirographs produce is not quite hypocycloids, we're not following the rim of this little circle that's rolling around with somewhere inside. Just inside, and that's where you put your pencil through. And so, actually what these spirographs produce are hypotrochoids. So you can go and tell people that because then you sound awfully knowledgeable. Actually I think we'll find it's a hypotrochoid, that's how to make friends, right? So these are lovely toys, and you can produce some very pretty pictures. What I want to talk about though is epicycloids, especially, so the picture at the very start of the lecture was an example of a cardioid. And I just wanted to give you a nice couple of facts about these arches 'cause we talked about what happens with the cycloid. And know that the area under the cycloid arch is exactly three times that of the circle that produces it. And Christopher Wren proved that the length of a cycloid arch is exactly eight times the radius. And you can actually derive formulae for these epicycloids as well. So here, this is the nephroid, right? It's produced by rolling a circle of radius R, let's say along one of radius two R. In general, if you have a rolling circle of radius R, and let's think about what happens when you roll it along a bigger one that has radius K times R for sum K. So for the nephroid, K equals two, for the cardioid, K equals one. For this guy, which I don't think it has a name, but it is an epicycloid with five arches, K equals five in that case. So the number of arches is the amount, the multiple of the radius with the biggest circle. So there are some facts about the area under the arches and the length. So for a cardiod, so we are looking at the area kind of between the fixed inside circle and the curve. And the cardioid, I guess, is kind of one arch really wrapped around. And it can be shown that the length of the cardioid is 16 R, 16 times the original radius, and there are nice whole numbers and the shaded area, so the area between the fixed circle and the the cardioid turned out to be five pie R squared, five times the area of the rolling circle. If you want the whole cardiod, if you want to fill in that bit and just get the whole cardiod, you could add in another pie R squared if you want it. Alright, so there's a result in general that says, if you are rolling around something of radius K times the rolling thing, then there's this formula for arch length, eight R times something, one plus one over K, and C for the cardioid, K equal one, right? So one plus one is two. So that gives you your 16 R, yeah? And the area, so the bit that's in blue on that slide there, the area is pie R squared times something. And the something is three plus two over K. So again, cardioid K equals one, three plus two is five. So we retrieve the thing we already know. But you can work these out, and this is true for any these epicycloids where you're rolling along a circle a whole number multiple K times the radius. So we could work this out if we wanted to for this five. This thing looks a bit like a flower, and K is getting to be a bit bigger there. And so, if we were to work this out, one plus one over five, it's a bit more than one. So the arch length for each arch, it'd be eight R and a bit. And the area under each R would be K is big. So two over K is small, so it's three pie R squared and a bit, area under each arch. Now, what happens when K gets bigger and bigger and bigger? Well, then, what you're looking at is you're rolling a circle along a circle whose radius is getting larger and larger and larger and larger and larger and larger and larger until at the limit, it's a straight line, a circle of infinite radius, right? A straight line. So we ought to, it ought to be that we might hope when K gets very large and tends to infinity, one over K becomes very small and gets closer to zero. If we actually do that, the limiting case of these things should give us what we got for the cycloid. And that exactly does happen. If you put K, whose is infinity, or let K go to infinity, then one over K becomes essentially zero, and you retrieve eight R for the arch length, exactly what Wren found. If you, in the area formula, if you let K get very big, two over K becomes close and close to zero, and in the limit, that formula becomes three pie R squared. That's exactly what we've got for the cycloid. That's really nice. In the limiting case of an epicycloid, you get back to being a cycloid. There's a really nice way to produce, oh, there it is, a nice way to produce a cardiod. I won't go into the details, but you can try this. You can try this at home. What you do is you get a circle, you mark off some points equally spaced around the circle. So I've got 12 points cut out like a clock. This is just as a starter. And what you do is you join each point to its double. So one goes to two, two goes to four, three goes to six, four goes to eight and you're sort of thinking, hang on, what does seven go to? There's no 14 on here, but it's kind of like the 24 hour clock. If you go around seven more from seven on an actual clock, you don't get 14.00 o'clock, you get to two o'clock. Right? So you just do that same thing in this situation. So every point's going to its double. When you do that with 12 points, you get something that looks not very nice, but let's double the number of points, and let's do it again. And let's do it again, and let's do it again, and let's do it one more time. And you can see, look, that's not a mathematical proof, but you can do a mathematical proof that what you do then to get here, kind of the involute of all these straight lines is a cardiod, and that's really pretty. Another place you might see what looks like a cardiod is in a cup of tea. You see on that picture, look at the light. It's sort of making a curved shape that looks like a cardiod. And well, that's interesting. Is it? Is it a cardioid? And what is this curve actually? So I've got this word on the slide, a caustic. If light hits a curved surface and reflects off, the curve produced by the kind of the boundary of the reflected rays is called a caustic. So we were asking, what is this caustic? And let's see what's going on. So imagine first, if you have a light actually on the circumference of a circle, and the rays are coming off it, and we know in physics that when light hits and reflects off a surface, the angle that it comes in at is equal to the angle that it comes out at. Angle of incidence equals angle of reflection. You might remember that. So we've got these two equal angles here. And imagine now, I'm interested in what's the reflected ray, what are all those reflected rays doing? So imagine that the lights is at zero or the point 0, and it hits the circle at point p. So I've just marked off some end points around the circle. So as it hits at the point p, what is this q? What's this q? Well, there's a lot of symmetry in the situation. And actually, because as these angles are the same, it means that chords have the same length. And then the bit of the circle they mark off is the same length. So the distance from zero to p must equal the distance around the circle from p to q. So actually that's got to be p again. So q must equal to p. So this ray is going from p to two p. It's exactly this doubling thing that we saw gave us a cardioid. The problem is, the light source probably isn't on the rim of your cup of tea. It's probably somewhere in the distance. So maybe we want to think about when light is coming from a far distance and bouncing off then the circle. And normally if that's happening, we sort of assume that the light rays are then parallel'cause they're coming from infinity. So if you solve that problem, it's a bit of a complicated diagram. There's too much going on in this diagram. Let's get rid of these parallel lines. You just get this. It looks all right. Let's add a few more points, and then let's clear things up a bit and just show you very clearly what this curve is. Is this a cardioid? Looks cardioidy. I don't even quite see this, but here is a cardioid superimposed. It's all right in the middle, but at the bottom and the top there are these little tiny gaps. It's not quite right. The thing that is quite right is an nephroid. That is in fact what you get if you have light coming, parallel beams reflecting for circle, you get part of an nephroid. So in practice, what's happening in your cup of tea is that the light source is neither on the rim of the cup, neither is it at infinity. It's somewhere in between. And so, the curve you get is sort of between a cardioid and a nephroid very slightly depending on where the lighting is in the room you are in. So that's a nice way to see these things. Where else can you see cardioids? If you're a sound engineer, you'll know what that is. It's a cardioid microphone. You don't see that thing, but that's the pattern of where sound is picked up by such a microphone. It's very useful if you want to pick up just the stuff that's in front of the microphone and not the extraneous noise behind. If you're recording live music or something, you want to hear the singer's voice, but you don't want to pick up the audience sound. The shape there is cardioid sort of rotated around that point One other place, the Manga on set. So this is probably the most famous flat in the world, right? This largest black area is a cardioid. Now, without going into the whole geometry of fractals, which we don't have time to do, I could at least maybe convince you that this is plausible. This picture is drawn on a plane, but it's using the geometry of what called complex numbers, which if you've heard of them, fine, and if you haven't, also fine. But what you need to know about this is, when you produce this design, the way you do it, it's a sort of iterative process. You take a number, you square it at a constant, and then you repeat that square at a constant, repeat, repeat, repeat. If when you do that, the answer gets bigger and bigger and bigger, and sort of zooms off to infinity, that's the blue bit. If instead it stays kind of where it is and converges to a finite amount, that's the black bit. So the other bit you need to know for this is that these kind of numbers we're working with complex numbers, you can put them, you can draw 'em on a picture by saying how far away they are from the origin and what angle the line to them makes with the axis. When you square one of these things, you square the distance from the origin and you double the angle. And so, it's at least plausible, I hope, that something which involves angle doubling might be related to the cardioid'cause we've already seen, you can make that with doublings. Okay, so that's that's cardioids. I want to spend five minutes now on talking about involutes, in particular, circle involutes, because they are some nice applications. And the first one is to do with gears. So as in, clockwork toys or other kind of mechanical devices. So when you have a system of gears or cog wheels, like you might find in a clockwork toy, what you want is you have gears that are transferring energy from one to another, and the input gear is kind of rotating at, usually, at some constant speed. And it's interlocking with another gear, and it's transferring the energy across, and what you want is, you want the teeth to be in constant contact so that the energy is transferred efficiently. And if the first gear is moving at a constant speed, then for most applications, what you want is the second gear, assuming they're both circular gears. You want that to be moving at a constant speed, not the same constant speed if they're different sizes, but you want to preserve that, and transfer the energy as efficiently as possible. Now, if it turns out, on the left in this slide, there's a picture of an involute of a circle, just to remind you, it looks a bit spiraly. And on the right, is a picture of two gear wheels with their interlocking teeth, and one is transferring energy to the other. Now, if the profile of those gear teeth is an involute of a circle, then it turns out that at the point where they're touching, they're constantly touching, the point where they're touching at any time, if we think how we make these circle involutes, we think about unwinding this line to a thread, unwinding it from a circle, that is basically a tangent to the circle the whole time. And so, if both of these things are involutes of the circle, then whenever they touch, the force, the energy is transferred along a line that is tangent to both of the circles. And that makes for a very smooth transfer of the energy. So these are really great for gears, for the profiles of those gear teeth. Another brilliant advantage of circle involutes in this context, and they used to use cycloids actually, cycloid profiles, but involutes are better for most applications because of a practical reason. It turns out that as long as long as you keep the same, what's called diametrical pitch, which is the number of teeth per inch on the wheel, on the gear, then it turns out you can use identical teeth on any size gear. So that means, if you're producing these things industrially, you don't need to have multiple different possibilities. You can use the same on every gear. So that's a very useful property for them to have. So, yes, you see archs of involutes, bits of involutes of circles in gears. The other application is inside a nuclear reactor. So actually what I'm talking about is, I think I said nuclear power plants earlier. This isn't quite that. It's nuclear reactor still, but, so the pitch you see is of the Oak Ridge National Laboratory in Tennessee, in America. And there they have the high flux isotope reactor, and they're not interested in making energy. What they're interested in doing is producing heavy isotopes of elements for various scientific applications. And what you do, there's a nice video about this on YouTube, but I'll put a link in the transcript, but I've just got a couple of pictures from it. What you do in in a nuclear reaction and what they're trying to do is to make heavy isotopes. So they want to bombard lighter isotopes with neutrons, and then they become heavier, right? Simple guide to nuclear actions. So what you do is you have your fuel, which is uranium oxide, and you don't want to have too much of it together in one place 'cause that's not good. But inside the reactor core, inside these cylindrical containers, you have fuel strips and they are the fuel, the uranium oxide, sort of sandwiched into aluminium strips. And then you have a whole bunch of those, and you want to put them into your cylindrical core in such a way that we don't want them touching because that's not good. And we need to leave a little bit of space between them, and the same amount of space, constant amount of space, because you really need to control very much the amount of heat. And between them is the coolant. The water is going between them constantly cooling them down because there's a lot of heat generated. There are a million billion neutrons per square centimeter being produced, right? But hat makes for a lot of heat, so the coolant needs to be able to pass equally between all of these things. So you've got these rectangles and you've got to fit them into a cylinder in such a way around a circular center in such a way that they have constant distance apart. So this is the kind of thing you you are trying to do. So what do you need for that? Well, it turns out that circle involutes have this brilliant property. Here's some. If you put them equally spaced around a circle, then those curves stay equally spaced forever, right? As long as you want to extend them for, they remain equally spaced. So this is it, I quite like this picture. They remain equally spaced. This would not work for any other curve. We tried archs of circles, and they would end up meeting, getting closer together. So involutes of circles precisely solve this problem. And this is a mock up of one of these insides of the nuclear reactor. And you can perhaps see all these beautiful involutes just radiating out from the central circle. So that's obviously very real application of involutes. And we've really only scratched the surface of what we could say about roulette curves today. I'll have to tell you another time what happens if you roll in ellipse along a sin curve, for example, but segue, if you are interested in ellipses, then you should come to my next lecture, which is on The Surprising Uses of Conic Sections. The ellipse is one example of the curve you can get when you slice through a cone at different angles. Another example is a parabola, which is shown in this waterfall here. So I hope you've enjoyed today's talk. Next time you have a coffee break, look out for the cardioids, and we will stop there. Thank you.(audience clapping)- There was a question about the competitions that you were talking about at the beginning. Are there any thoughts on what mathematicians were doing at the time or prior outside of Europe on these problems? It seems a bit European, the emphasis.- Oh, yeah. You're right. I would have to say, yet, I'm not sure. I think I would want to look at certainly what was happening in China at this time because I know that the Chinese mathematicians certainly developed something very like Cavalieri's principle. I mean, it's actually the same thing, and they use it to come up with a really fantastic way of deriving the volume of a sphere by slicing these shapes. You started with a third of a cube, and then you can find the area that the two cylindrical tunnels intersect. You can find the volume of that thing, and then you use that in a way, if I had 15 minutes, I could tell you to find the volume of a sphere. And it's this idea of slicing, but in three dimensions. So they certainly had principles like that perhaps before Cavalieri's principle, I'll have to check. And so, yeah, that would be my first place that I would go and look, what was Chinese mathematics doing at this time? Yeah.- Are the arches produced by cycloids used in bridge construction and things like that? And I'm thinking of some of the rather essetary arches that in Isambard kingdom, Brunel used to build bridges.- So without having knowledge of every single one of those bridges, normally, what you find in bridges is catenary, which is a slightly different kind of curve. And that's the curve formed if you take a chain of metal and just let it hang naturally under gravity. It forms this curve called a catenary, and quite often in bridges, that's the curve you get. So I would say probably catenaries, but I don't absolutely, be sure. Yeah.- Do we know the equation of different cycloids if we were to go after or something?- Yeah, so yes, we know the equation of cycloid and I didn't write it down, but it's very easy to find online or I could tell you afterwards. So it involves sins and coses basically. And you can give a, what's called a parametric expression. So you can give the X coordinate in terms of cos theater and the Y coordinate in terms of sin theater in some way that I didn't quite write down, but yeah, it's easy to find online. Yeah, Galileo, though, those people didn't have that information because it was even before we really had the concept of the equation of a curve. So, he actually was doing something fairly sensible, just doing a physical experiment like that because there just wasn't a mathematical machinery at that stage to be able to write down the equations that we now can use to do all this stuff with the techniques available to us now.- I'm going to bring it to a close there. Thank you to our audience for joining us today, and for your attention online. And please join me in thanking Professor Hart one more time.(audience clapping)