- Proportion has long been central to our ideas of form and beauty, and today I want to talk to you about the mathematics of proportion. And along the way, we're going to find out what links honeybees to the architecture of Le Corbusier. Why is A4 paper like this exactly 297 millimeters long? And what can an ancient Egypt scribe tell us about baking cakes? So let's begin at the beginning, and this diagram here, Leonardo DaVinci's famous Vitruvian man named for the Roman architect Vitruvius. Greek and Roman thinkers were absolutely convinced that proportion was key to beauty. Aristotle said so, and along with symmetry, which he also said was very important, and he said it's mathematics that helps us understand these things. Of course, I agree. Vitruvius said that the human body is put together with fixed proportions, each part corresponds to the whole in certain fixed, beautiful proportions, and he said that therefore, the ideal building should also have fixed proportions, and that would make it beautiful. But what do we mean by proportion? Well, one of the reasons why Vitruvius was so influential, and throughout Western art architecture, you know, it's a name that you hear over and over again, in part, the translations of his work from Latin into languages like French and English in the 16th and 17th centuries helped his work to become more widespread. And one translation by Claude Piroux in 17th century described proportion by saying it's all about relation, so the relation, for example, that windows of eight foot high have with windows of six foot high, when the one of four foot wide and the other are three foot wide. In other words, it's eight to four being the same as six to three. It's proportion, it's ratio that we're talking about here. So that's what I want to explain more about, and explore today, is this idea of ratio and proportion. And we're going to begin with probably the most famous ratio of all, the so-called golden ratio. Now, you read about the golden ratio, not by that name, in Greek texts like the work of Euclid, most famous geometry textbook of all time, I should think. In Euclid, often you encounter this ratio when we are talking about regular shapes, regular polygons, the Platonic solids, things like this. So what's a regular polygon? It's a shape where, you know, it's made of straight lines, and all the lines are equal, and all the angles are equal, so square, equilateral triangle, that kind of thing. Now, if you are an ancient Greek geometer, what you want to do, if you have a shape that you like, you want to be able to construct it using just a strange edge and compasses. So you've go to these geometric constructions, and you may have learned to do things like perpendicular bisectors, and equilateral triangles when you're at school doing geometry. But how do you draw a regular pentagon? Well, without going into details of how you do that, it is possible, no guarantee it would be possible, but it is possible, but if you're going to do this, let's start with, you know, an edge, a side A B, let's say that has unit length one. How are you going to construct the next vertex, vertex C? Well, if you happen to know how long the diagonal is, and I've just given it a name X there, if you happen to know that length, then what you could do is you could take a circle, center it at A, radius X, that's definitely going to go through your point C, and then you could take another circle, center it at B, radius one. That will definitely also go through the point C, so they'll intersect at a point that can be the next vertex of your pentagon, right? I mean, actually isn't quite the way you do it in real life, but that would definitely allow you to do this construction. So it depends on knowing the length of this diagonal. So can we find the length of diagonal of a regular pentagon? You can, with through the wonders of triangles, which are the best things, here's a pentagon, and I've drawn in all the diagonals, so it's got this kind of pentagram design in it. And whenever two diagonals cross each other, they each cut each other into a long bit, technical term, and a short bit. I've put the long bit in a sort of pale green color, and the short bit is an orange color. And we are going to find the whole length X by finding out the length of the long bit, and the length of the short bit, and adding them together. And it's going to be three pairs of triangles here, so if it's been a while since you've done geometry, you can just sort of daydream for a little, for a minute, but don't really, though,'cause it's very, very fun and exciting. So diagonal is the long bit and the short bit, and here's two triangles. They're the same triangle. They're congruent to each other. How do I know this? Well, they're both isosceles right? They've both got two equal sides. And the angle between those two equal sides is the same, because it's the angle between diagonals of a regular pentagon. For the blue triangle, it's the small, regular pentagon, and for the red one, it's the big pentagon, so isosceles with the same angle between the equal sides, and they share the base, so those things are congruent triangles. So in particular, that tells us that there are two sides that I've marked there with a little line, those lines are equal in length. What is that line? It's this short bit of a diagonal. So we've got that, okay. Now for two more triangles. They don't look like triangles,'cause one of them's on top of the other, but there's a green triangle here, which is the whole height of the pentagon, and it's an isosceles triangle with the two sloped sides are equal there. They're the diagonals of the pentagon, and then you've got the base, which is a side of the pentagon, that's the green triangle, and then the yellow one is similar to it. They're not congruent, but they're similar triangles,'cause again, it's isosceles. It's got to the two sloped sides, though, this time with the long bit of the diagonal, and then the base is the short bit of the diagonal. So those are similar triangles. What does that tell us? It tells us that the ratio of corresponding sides is the same. So if you take the green triangle, it's short, it's base, it's just the side of the pentagon, it's one, and the slope side is X, so that one to X is the same as the base of the yellow triangle, which is the short bit of the diagonal, to the long bit of diagonal. So you've got this ratio going on. One to X is the same as short to long. So the division of the diagonal has this nice property that the proportion of it it's the same as the side to the whole diagonal. We haven't found them yet, but we're about to. Here's our final pair of triangles, so a purple and orange triangle there. Again, they are the same as each other. They are, they share their base, and they're both isosceles, and they've both got the internal angle of a pentagon as their top angle there, opposite the base. So those are congruent triangles, and so all of the corresponding sides match up. So those two sides that are edges on the orange triangle, they're sides of the pentagon, those have length one,'cause it's a pentagon with side one, and so that means the sides on that purple triangle, which are the long bit of the diagonal, they are also length one. So we know now that the long bit of the diagonal is just length one, and the short bit, well, that's got to be X minus one,'cause the whole thing is X. So what could we do now? We now can get a relationship between an equation involving X,'cause this one to X equals short to long gives us this, turn it into a fraction, one over X equals X minus one over one. And if you rearrange that a bit to multiply on both sides by X, for example, look, there's a quadratic equation, and we can solve quadratic equations if we remember, you know how to do that, but that is a solvable problem. And if you solve that problem, you get an answer, that X is a half times root five plus one, which is about 1.6, one eight-ish... It's an irrational number. There's no fraction that it is this exactly, but approximately 1.618. And that we now call phi we call that, it's a golden number, so-called, and the golden ratio is one to phi. So that is the length of the diagonal of a regular pentagon, and that's where you see it in, for example, in Euclid. We've got a nice property here that this line, this number, the sort of X here, or we're now calling it phi,'cause that's the more common name for the golden number. When you subdivide it into a long bit and a short bit in the way we have, those numbers are related in the same way that the long bit is related to the whole thing, so one to X is just up here. We can say it now in terms of this phi. One to phi is the same as phi minus one to one. So that's a kind of nice equality of ratios there. That allows us to produce something called a golden rectangle. Now, this is supposed to be the, often claimed to be the most beautiful rectangle there is. I'm not sure I necessarily believe that. I think there have been various tests done, you know, experiments, which, tick which rectangle you find the most beautiful, and golden rectangles don't necessarily win such competitions. However, it does have a lovely property. If you take a rectangle with sides in the proportion one to phi, and you slice a square out of it, get rid of a square from that, then what's left, that rectangle that's left over, well, its sides are going to be, its short side is going to be phi minus one, and it's long side is going to be one, and we know, because I've put a little golden crib sheet up in the top right hand side to remind us what we know about phi. We know that one to phi, so the proportions of the original rectangle, is equal to phi minus one to one. So this smaller rectangle we've produced is exactly the same shape as the original one. You've got another smaller golden rectangle, and you could keep cutting squares out of this, and each time you get another smaller golden rectangle, you keep going forever and ever, so it's other beautiful property of this rectangle. Okay, so that sets a few things about this wonderful golden number, or golden ratio. The golden number is phi, and the golden ratio is one to phi, so yeah, a number on its own isn't a ratio. I want to tell you about the link to the Fibonacci sequence, because that's another really cool property of this golden number. I've put Fibonacci in quotation marks, because it's not Fibonacci sequence. Like, he didn't invent it, he popularized it. He put it in a book that he'd wrote, and with a, rather dare I say it, slightly fatuous example about rabbits' populations. They don't really, you know, that's not really what happens with rabbit populations, however, he brought it, he popularized it. But this sequence had been known, you know, in India, and other places for hundreds of years before Fibonacci, however, this is what it's known as, so I thought I should at least say that word. What is the sequence? Well, I've put the first few terms here. You may or may not have seen it before, that the sequence rule is to make the next term, you just add the previous two. So if you look, there's an eight up there, eight is made from adding three and five, and then five plus eight is 13, eight plus 13 is 21, and so on. So the rule is, if you wanted a formula for it, you get the N plus first term, F N plus one, you take FN, the Nth term, and then add the N minus first term. So that's the rule for making the Fibonacci sequence, start with two ones, and then you just turn the handle, and keep going. What's that got to do with the golden number? Well, if you take the ratios of consecutive terms, so you find out one over one, two over one, three over two, five over three, and so on, so you work out all of these ratios. You don't have to do it right now in your head, but I've started to put what this is, you know, underneath them as a decimal. I've stopped at three decimal places just, you know,'cause of space, but you can see what starts to happen. It moves around a little bit at the beginning, but it starts to settle down, and by the sort of seventh or eighth term, you are getting 1.618, 1.618, 1.618, and a few, you know, infinitely many terms after that, but I've put those. So it looks very much as if this is settling towards some limit, as you know, your terms, the number of terms get higher and higher and higher, you're getting closer and closer, it looks like, to a number which seems to resemble our phi, our golden number. So why might this be, why might this be? Well, if it does, if this is going to end up at sort of getting closer and closer and closer to some limit, which is what it looks like, if that's true, then it means once you get large enough, every time you do this, you are very close to whatever this number is. Let's call it L for the moment. We don't want to jump to any conclusions. So you've got sort of the N plus first term, FN plus one over FN. It's roughly the same as FN over FN minus one, which is roughly this L, whatever it is we're hoping to find. So now all we need to do is to take that equation that gives us, or that expression that gives us, you know, the next term in the sequence. If I divide that through by FN, you'll see what happens, I've done it there, then what you end up with is, right, this ratio on the left, you've got FN plus one over FN. That's equal to our L, assuming approximately, then FN over FN, well, that's just one, and then FN minus one over FN, so that's that L sort of the other way up, it's one over L. So we get this expression, L is equal to one plus one over L, and that, I'll slightly rearrange it, just so that I can show you it's exactly the same as the expression one over phi equals phi minus one that we know about the golden ratio, the golden number. So one over L equal L minus one. So they, these two things satisfy the exact same equation, and so that's why when you solve that equation, you get the same answer that comes out. And it's true of the Fibonacci sequence. If you take the ratios of consecutive terms, it does end up closer and closer to the golden number. So that's a really nice fact about the Fibonacci sequence, and about the golden number. We hear a lot about the golden ratio, and how it's supposed to be, you know, everywhere in nature, and you know, the human body. A beautiful human face is supposed to have loads of golden ratios in it, and golden, you know, everywhere. Let's think about that a little bit, and where did that come from? Part of where that comes from, that belief came from is not really the fault of Luca Pacioli, who wrote a book in 1509 called Divina Proportione about the golden proportion, the golden ratio, but this is the start to the trouble, I would say. So he wrote this book. This is hundreds of years before the word golden ratio was, the phrase was invented, but he though phi was amazing and wonderful. He wrote this book called The Divine Proportion, which a lot of it was about the golden number, and he gave lots of properties of it and said, you know, it's fantastic. 13 properties he gave. That's not cause 13 is a Fibonacci number, but because he was trying to connect it in all the ways he could think of to the divine, and in particular, the Christian God, and there were 13 people at the last supper, okay? So therefore, he'd done 13 properties. I'm not sure which one is Judas, I don't know. But what else did he say? It's like the divine. Why is it? So one reason is he said, you know, I said earlier, you couldn't represent it as a fraction, so it's not that it's irrational. I mean, it is irrational, but that's not, we're not saying that about the divine. What we're saying is you can never fully know it. You can never fully know it, because you can never fully write it down as a fraction. It goes on forever, and so does God, and you can't fully know God. What else? When you define the golden ratio, you do it in terms of this, you know, the small part to the large part is the same ratio as the large part to the whole. It's defined in terms of three quantities that are all related to each other, and so that's a bit like the Holy Trinity, he says. So there, and then he gives lots of wonderful geometrical properties that I'm more qualified to know what I think of, and illustrated here is one of those. So you've got here that the red lines are showing you one of the Platonic solids of the five Platonic solids, the icosahedron. It's made of 20 equilateral triangles, and the amazing, this is a wonderful thing. You can put the vertices of this into groups of four that make up three mutually perpendicular golden rectangles. And they're sitting inside those 12 vertices, make up three groups of four vertices that define these golden rectangles that are sitting at all the right angles for each other in the middle of the icosahedron, which is really nice. So Luca Pacioli gives lots of properties. He said this is a wonderful thing, the divine proportion, it's amazing. Now, also in his book, in this same book, there are sections about architecture, and so of course, naturally you might think,"Well, so he's going to say how the divine proportion, how it's involved in architecture," but no references at all to the golden ratio in those sections. It's just not mentioned at all, which is curious. So I have to think,"Oh, okay, how, why do?" You can sort of see why it might have been thought that they would, but if you look at the book, it's not in there at all in the sections on architecture. And these words, golden ratio, golden number, phi, that symbol, these are first used, not by Pacioli, not by Vitruvius, not by Leonardo, but in the late 19th century by a chap called Adolf Zeising, who wrote a book saying, you know, this is an amazing thing, and it's all over nature, and art, and architecture. It's, I would say it's sort of a bit in these things, but perhaps not to the extent that he was claiming. For example, that phi, why is it, why did he call it phi? That's the first letter in the name Phidias, the architect who designed the Parthenon, but there's no record of any, you know, anyone writing that we are going to use this particular ratio in the design. So let's have a little look, then, at architecture. Is the golden ratio used in architecture? If it is, where is it used? So there are lots and lots of ratios and proportions in architecture. It's absolutely critical and important to our understanding of form and design, and, you know, architects are constantly choosing ratios such that they can get, you know, nicely shaped buildings, but not so much the golden ratio, certainly not in the work we talked about Vitruvius, the Roman architect. He wrote in De Architectura, he wrote a kind of manual for how to build all the possible kinds of temple you might want to build, for example, and very, very detailed instructions. You know, he was absolutely clear that there are right ways and wrong ways. (chuckles) Here are the rules you need. Here are the proportions of the, you know, six different kinds of temple, or whatever. This is how many columns wide. This is how many columns deep they should be, and this is the gap between each column. So not about how, you know, your column should be three cubits wide, or whatever. It's what your column width is, and then the gap between the columns in terms of the column width, and then how many of the columns you have, and so you can work out. He doesn't explicitly say that, you know, the Eustyle temple is in this ratio of rectangle, but he explains what the columns, how many columns, and how wide, how the gaps should be. And you look out what the ratios then are for the rectangles, the footprints of these temples, and they're not, none of them is golden ratio, I'm pretty sure. So I've worked out a few of them. Didn't find any golden ratios. I think the issue is, and it's the same sort of thing that happens with the claims about the human face. There are lots of numbers that you can, that you could measure. There are lots of measurements you could take of a temple. And if you take enough measurements, and you're willing enough to approximate and say,"Oh, that's roughly 1.618-ish," you know,"That 1.7 was nearly 1.618," then you can find golden ratios, but there's not really, I would say, enough evidence that the Greeks and Roman architects were consciously using this wonderful ratio. They had other wonderful ratios. So that's the state of play, I would say, with the Greek and Roman architecture. So is there any use of the golden ratio in architecture? Yes, there is. Yes, it was, luckily. Yes, there is. Never ask a question like that if the answer isn't yes. Yes, I want to give you probably the most important example of its use, and that's by the Swiss architect Le Corbusier. This is a commemorative coin produced by the Swiss mint, representing his Modulor system, which I want to talk to you about. So Le Corbusier, he is well known, for good or ill, for having a very firm belief system about what architecture should be, let's say. Notoriously wanted to not down half of Paris, and replace what had been there with 60-story tower blocks, because that was efficient, and that was the right, you know, that was the thing you should do with the wonderful new building materials. So he was able to take advantage of some new building materials, like reinforced concrete, to have more freedom to design structures and buildings. No longer did you have to have a wall there, because otherwise the house falls down. You could have it as open plan as you wanted to, and you could put walls where they were best to serve the people who were using or inhabiting that building. So he famously said a house is a machine for living in. That's what it should be. That's, we're making this machine to serve this purpose of being inhabited by humans, and so it shouldn't have stuff that isn't important to that. It should have precisely the things it needs. So he was all sort of, you know, let's be efficient. Let's almost like let's, you know, we've got the model T for cars at that time, you know, let's have a production line for houses, and let's make them, let's think of them in this way. He wasn't too keen on ornament, and things like this, decor, no. He said art was all right, but just sort of have nice, little, fancy, decorative touches, and, you know, Corinthian columns and things. No, we've got this for living in. So one of his frustrations, or the things he was unhappy about was the system of measurements that was included in the metric system, and it's because it's not really to do with human scales. The meter, unlike so many other unit of measurements that we had, have had at other times, it's nothing to do with the human body and human scales. The meter is defined in terms of, you know, millionths of the circumference of the earth, which okay, fine, but you know, if you're a human, actually, a foot or, you know, even a cubit, something like that that's defined in terms of the human body, feels, to him, more natural. And what he wanted to do was to take that idea of having human scales, things that are relevant to humans, and to create, as he said, a range of harmonious measurements to suit the human scale, universally applicable to architecture and to mechanics. So the idea is you've got this scale of measurements that works with humans. You know, these are the things we're building the buildings for, from the smallest scale, from, you know, like, the door handle in your home, to the furniture, to the room, to the house, to the city, all of those scales, there should be something in this range of measurements that will suit, and then they'll all harmoniously work with each other, and everything will be beautiful and wonderful. So it's all very good. So what is this scale he's got, this Modulor system? So it's based on the height of a man, 1.83 meters, and then with a raised arm, so that brings the total up to 2.23 meters, which is 89 inches. 89 is, I'm sure you've spotted, a Fibonacci number, so that's where that comes from. What about this 1.83 meters? Well, if you are, if you're good at converting between meters, and feet, and inches, you might know what that is. Corbusier said he chose that height for the man, because in English novels, the good-looking men, who are usually policemen, are six foot tall. So I don't know how to deconstruct that, (chuckles) but anyway, that's what he said. And so 1.83 meters is six foot tall. So we can have a discussion about whether, how great or otherwise it might be to put your whole design based on a man who is six foot tall, but there we are. That's where it came from. And you can see, I mean, you can see in the design there are things that look like golden rectangles, and they are golden rectangles. So you take his whole height, halfway up is the navel, and then if you go right to the top of upstretched arm, that's a golden rectangle. But up to the point of the top of the head, that's a square. So if you take a square out, you get a rectangle left, a golden rectangle left. And so then there's all this whole range of golden rectangles getting smaller and smaller. You can get them bigger and bigger, and combined with squares, they all mix in beautifully together, and you get this range of decreasing and increasing measurements that you can use. I want to show you just a couple of the buildings or interiors that he designed using this system. And I will mention Modulor, that name, module, and or, the French for gold. So that's, the clue is in the name. The golden ratio is everywhere, and golden rectangles are everywhere. So this is an example. This is an apartment block in Marseille, one of several that he designed, and you can, you know, just have a look, spot some golden rectangles, the tiles on the floor, on the ceiling. There's some windows that are golden rectangles. So there's a few just spot there, and then this is the kitchen of one of these, and you can look at those cushions, golden rectangles. You've got the windows are as well. The little kind of the grid shapes in that ventilation holes there, they're golden rectangles. You've got some squares in there, so the furniture as well, golden rectangles everywhere. So I think for me, you know, sort of pleasing to look at, and when you, whether or not you think golden rectangles themselves are particularly the most beautiful thing, it's still pleasing that you can have a system where everything aligns with everything else. You've got this shape repeated at big and small scales. You can create smaller and smaller, or bigger and bigger golden rectangles by taking squares away. So I think that's what helps to make it harmonious. It all fits together beautifully. So that's architecture. We also hear claims about the golden ratio in nature. Is the golden ratio everywhere in nature? Well, again, I'd say ratios are everywhere in nature. Some of them are golden, some of them aren't, and that's okay. It's all still interesting, but I want to talk a little bit about where they may appear, where you do, perhaps, get the golden ratio in nature. And first I will just say, you know, I've mentioned about the human face, and how it's supposed to have lots of golden rectangles or golden ratios in it. And just like I said about buildings, there are lots of measurements you could take of the human face, and there are lots of human faces, and who gets to decide what's beautiful and what isn't? And who gets to decide which measurements are the most important, and which aren't? So one, just one tiny example I found. There's a claim that in the, you know, in a beautiful face, the distance, the ratio of the distance from the center of your nose to the start of your eye, that distance, compared to the width of your eye, that's supposed to be the golden ratio, one to phi. So the width of your eye is supposed to be about 1.618 times the distance from the center of your nose to the start of your eye. Okay, I didn't measure my own face. You know I don't want to set myself up for failure. Maybe it was the golden ratio, I dunno. Instead, I looked at Audrey Hepburn's face. I thought, if it doesn't work for her, then it doesn't work. And for her, that measurement, that ratio was one to 2.3. So I thought, okay, it's not true. (chuckles) And this is, you know, I that's, that's an N equals one example, but, but there are so many measurements you could take, right? That particular, even if that particular one is about 1.6, what about, you know, the ratio of the height of your ear to the distance from your chin to your eye or something? You know, you could take so many hundreds of different things, and check them, and yes, some of them are going to be about 1.618, but some of them are going to be completely different. So I think, you know, we got to be careful about what is claimed here. But let's talk about spirals, because you do get lovely spirals in nature, and sometimes people talk about a Fibonacci spiral, or a golden spiral, so what's that? So here's how to make a Fibonacci spiral. You start off with the Fibonacci sequence, 1, 1, 2, 3, 5, 8, and so on, and you start, I'm going to point to the, you start here with the little red square. That has side one, so you are taking a bunch of squares whose side is, you know, the next Fibonacci number. Put next to it, another square of side 1, 1, 1, and then two, so you add a square of side two, and you're kind of going around 90 degrees each time, and adding the next square with a side the next Fibonacci number. And the reason it all fits together nicely without gaps is because the sum of the last two Fibonacci numbers equals the next Fibonacci number. So if you want to, you know, that square of side three, that you can see there goes next to a square of side two and a square side one, and two plus one is three, so there's no gaps, and it all fits together nicely. Where's the spiral coming in? Well, in each of these squares, you just put a quarter of a circle, like this, and that gives you something that looks quite spirally. It's not exactly a spiral, because it is just made of a bunch of quarter circles bolted together, so the curvature is changing kind of suddenly. It's not changing by much, so it looks all right, but it's not quite a spiral. Let me tell you what is a spiral. Here's two examples of spirals. There are different kinds of spiral. The one of the left is called an Archimedean spiral. In that kind of spiral, so in each one, you're kind of, you're gradually increasing your angle, and when the angle increases, you get further away from the center. This one is called an Archimedean spiral, and the distance from the center is going up in multiples. So every time you do another full revolution, you kind of add another multiple of whatever, some constant you might have. So the one I've got here, once you go one revolution around your distance one from the center, and two revolutions, two, distance two, three revolutions, your distance three, and so it's sort of adding up like that each time by some multiple of the angle. But this other kind of spiral are on the right here, it's more interesting, and it's more like the kind of thing you see in nature, because here, it's not about multiples, it's about powers, not quite powers of the angle that I actually wrote. That's not quite what I wanted to say, really, but it's powers that this is changing by. So this time, every new revolution you go by in this particular example, I've got what I call a two spiral, the distance is doubling after every revolution, these are called logarithmic spirals. The reason is that the log, the logarithm of the distance is dependent on the angle, right? So if you are, if you're doing A level maths, you might have done logarithms. And so this thing, what does it do for us? You can see the distances after every revolution you're starting at, well, at one point it's two, then the next time round, it's gone up to four, it's doubled, and then it doubles again to eight, and so on. Why is this more like what happens in nature? Well, it's because in a natural process, when something is being created or growing from a central point spiraling out, the matter that's there continues to expand and grow, even after it's been produced. So this is true for something like a spiral galaxy, where the galaxy is, you know, it's spinning very fast, and as the matter out from the center, it has still continued to expand, but also in something like a shell, it's growing from the center, but it doesn't suddenly stop growing, and just sort of sit there, like it would for an Archimedean spiral. So that is why you see, you do get these kind of logarithmic spirals, or spira mirabilis was the name given to them by the mathematician Bernoulli, who did a lot of research on them. Sadly, on Bernoulli's gravestone, I dunno if people know, on Bernoulli's gravestone, they were like,"Oh, he did some stuff on spirals, put a spiral on," and unfortunately, they put an Archimedean spiral.(chuckles) They put the wrong spiral on his gravestone. So, sorry, Bernoulli, that was done wrong. Anyway, these are beautiful spirals. They do look a bit like our golden spiral, but what kind of logarithmic spiral might be the best approximation to this, or what is this looking like? So we can, there's a bit of Fibonacci facts I need to tell you. If you think which, if we're trying to work out how far this, this spiral-like curve is from the center, then what you're doing is at each, after each kind of quarter turn, you're traveling across a square that's, you know, the side of a Fibonacci number. So if you sort of keep doing this, if you think of the Pythagorean theorem, the square of the distance is the sum of the squares on the side, right? So here, and you have to slightly believe me,'cause I'm going to say something a bit vague, but here, after kind of N quarter turns, roughly speaking, the square of the distance you've traveled is the sum of the first N squares of Fibonacci numbers,'cause those are the sides, right? So you want to know what's the sum of the squares of the first N Fibonacci numbers, and luckily for us, that is known, and it's just the product of the Nth and the N plus first Fibonacci number. So that's not obvious at all, and I'm not going to prove it to you right now, but this turns out to be true. So it's that number, actually, the square root of that, that we're interested in. That's the distance in the golden spiral after N quarter turns, so N of these sort of squares, and actually, if we want to account, you know, with our logarithmic spiral, it's how much you've multiplied the distance by out that you're interested in,'cause you're multiplying by that amount each time. So you want the ratio of this thing, compared to the previous term, or yes, and when you do that, we remember that our Fibonacci numbers, the ratio of consecutive terms is the golden ratio, approximately, so you actually do get phi coming out of this. So what this golden spiral does, or approximately, it approximately has a property that, after every quarter turn, you multiply the distance from the center by phi. So after every whole turn, you'd have done this four times, so you'd have multiplied not by two, like the example I had before, but by phi to the power of four, phi times phi, times phi, times phi. So actually what you're approximating is a logarithmic phi to the four spiral. That's what this golden spiral does. Now, the question is, why would anything in nature be interested in doing a phi to the four spiral specifically? And the answer is mostly they wouldn't. Mostly, the logarithmic spirals are there, and there are loads of them to be seen in natural phenomena, in shells, for example, but if you measure what number spiral it is, it's usually not that particular phi to the four one, and why would it be? There's a whole range, you know, in nautilus shells and others. There's a range of possibilities, and the ratio that they are bound by, you can find these things, and you know that these are interesting ratios in nature, not usually the golden ratio. You do sometimes, for real, see golden spirals, or those, you know, the analogs of those in algorithm spirals in some special situations. For example, in sunflowers, where the seeds are growing, you do see them, and the reason for that is actually because phi is an irrational number, it means after a whole number of rotations, you're very unlikely to sort of be overlapping something you've already had. So there's a good reason why you would, it's often the case that you do get phi in those kind of special situations. Most of the spirals in nature are not golden spirals, but just occasionally there are. There's something, some other ratios you might see in nature that I want to mention to you, and it has to do with populations. So there's a few proportions you could think about to do with populations. The first is predator-prey. So we all know you have fewer lions than zebras, for obvious reasons, and you know that there will be a certain proportion that seems to work well in a steady state. You could find and study that. Or what about adults to offspring? We know that, you know, a pair of ducks, a duck and a drake, I should say, will not lay precisely two eggs to reproduce the population. They will, there'll be more than two eggs,'cause very sadly, not all ducklings will survive to adulthood. So there'll be, again, there'll be a good amount. There'll be a proportion that you need to reproduce more than yourself in order to maintain a steady population. There's another statistic about populations that I want to talk about, though, and this is the proportion of males to females in a population. So in humans, roughly speaking, slightly more male babies are born than female babies, but by the time everyone reaches adulthood, it's approximately one to one, 50%, male, 50% female. And kind of a justification for this, why this might make sense, is about efficiency. Because if you think about, you know, all your ancestors, you know, you've got eight great grandparents, four male, four female, right? Half of your ancestors need to be male, and half need to be female, because to make a baby, hopefully not startling anyone here, you need one of each, right? So it makes sense, efficiency wise, to have approximately half a population male, and half female. But here's a pair of bees going out for little stroll. Maybe they're courting. They're different, because of something called haplodiploidy, which I hope I'm pronouncing correctly. It's a curious phenomenon with bees, where actually, not all bees need to have two parents. Male bees hatch from unfertilized eggs, so they only have a mother. Female bees hatch from fertilized eggs, so female bees have a mother and a father. So have a think like, what does that imply for the proportion of males to females in a bee colony? If the males only need a mother, and the females need a mother and a father, is it sort of going to be two thirds female? What's it going to be? So we can think about this, and we can draw the bee family tree, and we're kind of going backwards through the generations, right? We're looking at ancestors. So at the bottom, I've tried to, I've got very gender stereotypes. They're blue squares for males, and pink circle is female. I feel guilty now that I've done that, but anyway, it's just in case you can't see the letters. So your male bee has a mother only, then, so that's one generation back. Then you go back the next generation, the female has a father and a mother, oh, golly, (chuckles) father begins with F. Right, mother begins with M. (chuckles) A female parent and a male parent. And then going back again, the female bee will have two parents, and the male bee will just have a mother. And going back, and going back, you can keep going. But just have a look, count up how many there are on each generation back, 1, 1, 2, 3, 5, 8. That sounds like a familiar sequence. Why should this be the Fibonacci sequence? What is going on? Well, it's because there's the same sort of pattern. It's the same sort of rule for getting the numbers applies. So if we write something like A N to be the number of ancestors N generations back, so five generations back, which what I got at the top, there are eight ancestors, then we can work out the numbers of these. So that number will be comprised of a certain number of females, and a certain number of males. How many females are there N generations back? Well, everybody has a female parent, right? Everybody has a female parent. So the number of females in N generations back is just going to be the total number that they were in minus one generations back, that A N minus one. What about males? So how did you get to be a male in that generation? Well, you are the father of somebody, but only females have fathers, right? So if you're a male in generation N, then that, you are the father of a female bee in the previous generation. So the number of males in generation N is the number of females in generation N minus one, and that's the total number in generation N minus two. So if we put that together, we get exactly the same relationship. A N is A N minus one, plus A N minus two. That's exactly the Fibonacci relationship. So we started with one and one, and then we can just work out, and exactly the same rule gives you the Fibonacci numbers. So we get the Fibonacci numbers, which is amazing. And another amazing thing, if I asked about the proportions, so if you think, you know, what proportions of the ancestors are of each gender, and that might tell us what it is in the whole of the society, you exactly get, there are how many males in generation? N it's A N minus two. How many females? It's A N minus one. So the proportion, the ratio of males to females is the ratio of the N minus second Fibonacci number to the N minus first one, and as N gets bigger and bigger, this becomes the golden ratio, right? So the proportion of male bees to female bees in a bee population is the golden ratio, and that's lovely. That's lovely. So that's the place where, again, we do see the golden ratio in nature, and it's kind of an unexpected joy to see that. I'll talk about some other things now in the last 10 minutes, and it's proportions that we see around us in everyday life. So I said at the start head of piece of A4 paper, and I said, why it is A4 paper? It's fairly standard around most of the world, these sort of A series of paper sizes, A4, A3 is bigger, A5 is smaller, and so on, why is it exactly 297 millimeters long? And, you know, the first guess at this might be,"Well, maybe it was something in inches once, and this is their conversion. That's often the case." It's nothing to do with that, it's something else, and it starts with this chap, Georg Christoph Lichtenberg. He proposed that there's lots of different sizes of paper milling around, and you know, how are we going to get, what's the most efficient series of paper sizes? You know, we need different sizes for different purposes. What's the most efficient way of doing this? Well, what's very good in terms of efficiency is if you have a great big piece of paper, a folio, you ought to be able to make the next size down by just folding it in half and cutting, right? And then you can fold again, you can get octavo, quarter, and then you can fold again and get octavo, and so on. So if you do that by folding, and then you can cut, you're not wasting any paper. So this is kind of a good system, if these are your standard paper sizes. So that's what he suggested. So he said that what you want is the shape of your paper should be, have this property, that if you cut it in half, you get the same shape, but just smaller. It'll be half the area. So what do we need there? I mean, we could solve this now, mathematically. You've got your piece of paper, you cut it in half, and what you want is a rectangle that has the same proportions, the same ratio. So if your original one has side, let's say one, and the long side is X, then the one you create, now this time, it's long side is one, and it's short side is half X, and you want those to be in the same ratio. So you want the little rectangle short side over long side, half X over one, should equal short side over long side of the big rectangle, which is one over X, and we can solve this equation. It's easier than the Fibonacci one. We just get X squared equals two. Now, of course, that has two solutions. I'm not saying that the negative square root of two is not a solution of this, but we are dealing with pieces of paper here, so it's length can't be minus root two. So I feel okay in saying that X must be plus root two here. So X is root two. So that's the relationship, one to root two. That's what shape the paper should be. So for our A4 paper that's, if we want it to be 210 millimeters wide, then we know that the lengths should be squared to two times that, and that's 297, right? Give or take. So that's why, that tells you if you know it's 210 millimeters wide, then it must be 297 millimeters long, but that doesn't actually tell you why those particular numbers, right? The answer to that comes from the French Revolution, as I'm sure you were expecting me to say. There was a law passed. I love these French Revolution calendar dates, the 13th day of Brumaire sur le timbre, I think it's probably November sometime, in the seventh year of the Republic, and this was a law, and it was about taxes on various things, but they specified some paper sizes, a series of paper sizes of the Lichtenberg kind, with this exact, you know, the exact shapes, but they gave the sizes as well, and they were interested in them being fractions of a square meter, this lovely new metric thing that they'd invented. So the Moyen Papier is one eighth of a square meter in area. And this is what we do now, so our A paper series starts at A0. That's the biggest one you can get. It is, you know, by definition, the paper of this size, whose area is a square meter exactly. So that's A0, and then A1 is half that, A2 is a quarter, A3 is an eighth, A4, a 16th of a square meter. So once you know that, you can then exactly work out what it has to be. It's got to be a rectangle of that shape, with the area 1/16th of a meter. So if it's with this A, then the area will be A times the square of two times A, and that should equal a 16th of a meter. If you solve that, 210 millimeters. So that's why A4 paper, not only, that's why it's the shape it is, but that's why it's those exact dimensions. It's all about parts of meter. So we can thank the French Revolution for the shape and size of our paper. While I was thinking about, you know, that's kind of cool, that's two dimensional, could you do this in three dimensions? Like, could you have a cuboid that if you cut it in half, you get two cuboids the same? You can, you can solve that equation, but you know, let's think what cuboids we use in everyday life. I mean, I thought of bricks. You don't fold bricks in half. So that isn't what you want of a brick, right? That's, it's not this that we're interested in for bricks. You can solve that problem, and it's kind of fun. I encourage you to do that on the train the way home. But actually, if you think about something like bricks, you have a different, you want them to do different things. So a brick has a height, a length, and a width. What should those be? Or at least what should the proportions be? Brick walls, we walk past hundreds of them every day. As you'll know, if you've ever, I have never built a brick wall, I have to say, I have built Lego walls, and I know you have to alternate. You can't have all your weak points lining up, so you alternate them, and often you will do this by having a wall that's like, a full brick in depth, so it's sort of two brick widths deep, and that's the same as one brick length, and that means some of your, in some of your rows, or courses, they're called, the bricks can be facing this way, and in others, they can be just sort of length side on. So to do that, you need the widths to be half the length, or the length to be twice the width. You need that to do this. To give yourself a bit more play to make different designs of brick, there's another thing that would be nice. There's a third dimension here, the height, and I dunno if it's a good enough picture. It's just a brick wall on my road. The top row, or course of this, has got this slightly different finish, just to sort of say, this is top of the wall, slightly ornamental. It's got bricks that are on their side, and there are three of them that fit into the length of one brick. So this particular kind of course, to do that, you need the length to be three times the height. And so if you put those together, you get this nice collection of ratios. Height to width to length is two to three to six. And of course, you'll all be rushing home to measure a brick. Now, I will say, you've got to take off a little bit to allow for mortar. So you, you take, you decide what size your brick wants to be, and it'll be, you know, what you can hold in a hand, basically, but then it won't quite be two to three to six in the finished brick, but when you add in the 10 millimeters for mortar, that is the ratios that you're going to get. So that's bricks. That's something three dimensional. I want to finish in the last five minutes, I guess, by talking about everyone's favorite thing, cake. And in particular, you know, in baking, we've got circles here. We haven't really talked about circles enough. We always talk about circles in every mathematical talk, and about cake. If you are baking, there is so much mathematics in cooking. So much, it's wonderful. You've got volumes, and you've got, you know, if you're icing this thing, you've got, think about surface areas, how much icing do I make? If you've got some ornamental band or something, you need to know about the circumferences, or perimeters. There's a lot going on. The particular thing I want to talk about is cake tins, right? If you've ever baked, you will know that there's kind of standard sizes for cake tins. There are standard sizes, and they're usually inches, and there are standard shapes. You normally, cake tins are either round or square. You can get other fancy-shaped ones, but that's the main ones. And if you are working out how much do I need for my nine-inch-round cake tin? Then lots of cookbooks will have kind of tables that say,"You need this much for this, and this much for that," and my cookbook that I have at home, or one of them says,"Nine-inch-round tin requires the same amount of cake mixture as an eight-inch-square tin." It's like, that's interesting. Does it exactly say, is it? Because, of course, in there implicitly, they're making an approximation to pi, 'cause circles have something to do with pi, right? The area of a circle, pi R squared, and squares, you know, don't. So they're making some approximation to pi. Do I know any ancient, wonderful, historical sources of approximations to pi? And I did know. (chuckles) I did know one. This is the Rhind Papyrus, Rhind with an H. It's not about cooking. It's got lots and lots of mathematical problems on it. It's an ancient Egyptian papyrus, and I mean, you might be able to see the sort of nice pictures of triangles and things. But the problem I want to mention to you is problem 50, method of reckoning, a circular piece of land, diameter nine, what's its area? And there's an algorithm given. Subtract one ninth of it, and then square that, and you get the area. And there's other problems that do the same sort of thing, so we can deduce that the ancient Egyptian formula for the area of a circle of diameter D is find eight ninths of the diameter, and then square that. So that gives you 64, 80 firsts, 81ths, 64 over 81 times the diameter squared. Now we know what the area of a circle is for real, I mean, pi R squared, but if you're working with the diameter, pi over four, D squared. So what the ancient Egyptian scribe has done there is they've got an approximation for pi. I mean, they're not thinking,"I've got an approximation for pi," but in effect, they are using pi over four equals 64 over 81, or pi is 256 over 81, which is about 3.16. That's a pretty good approximation. So I thought, well, what does my cookbook, what's it's approximation to pi? So we've got the ancient Egyptian one. If I'm making a square cake or a round cake, my cookbook says nine inch round equals eight inch square. So by the way, these tins are, tend to be all the same height, so we don't have to worry about the height. That's just constant in there, H. So the area that you need, or the volume will be for a nine inch round tin if that's the diameter would be 81 over four pi H, and the, for the eight inch square tin, it'll be 64 H whatever the height is. So you can see what the approximation is. 81 over four pi equals 64, and low and behold, wow, pi again. So, you know, pi in a bit millennial later, my little cookbook in my kitchen uses exactly the same approximation to pi. So, you know, there's progress for you. So, but you know that it's, so this is a good approximation. I just really like this fact. I will spend two more minutes,'cause I can't resist telling you about just one final proportion puzzle which you can play with if you like. You know the old saying, you can't fit a square peg in a round hole. Well, you can, it's just there'll be gaps. So what I've got here, a square peg, on the left, a square peg in a round hole, on the right, maybe this is better, a round peg in a square hole. Right, what's better? If you have to fit one of them inside the other, what's better? And you can work out what's better, 'cause you can work out all the areas involved. So if the radius for this circle is R, both of them, then you get pi R squared. The big square there, it has a diamond, so two R, so it's area's four R squared. And what about the little square? It's a tiny bit more complicated, but actually look, see how it fits just exactly into the big square? It takes up half that area. So the square peg has area two R squared, and so you can work out what's better, square peg in round hole, you can work that out, and you get that it takes up about 64% of the area, but the round peg in the square hole, 79%, that's much better. So it's better, categorically, it's better to have a round peg in a square hole than a square peg in a round hole. That's very nice, but this is peak mathematician thing to do. In 1964, a mathematician called David Singmaster was thinking about this, and he said, "Well, that's all very well in two dimensions, but what happens in three dimensions?" What if you get whatever, you know, what's round and square and three dimensions? A sphere and a cube. What's the answer to this question in three dimensions? And you know, higher. What's better, square in round, or round in square? And you can work this out. We know the formula for volume of a sphere, and the volume of a cube. And it turns out that actually that, yeah, the cube only fills 37% of sphere, rubbish. The sphere fills, well still not very good, but it fills more. It fills 52% of the cube that it just sits inside. So again, round in square is better in three dimensions. And you can check it in four dimensions, and in five dimensions, round in square, round in square is better. Round in square is better. So now we know round in square is better in all dimensions, right? No, (chuckles) it swaps over nine dimensions. What? (chuckles) This is really, I just sort of wanted to leave you with this thing, actually, just a reminder. This is why we prove things in math, because we can have a really good idea of what's going on, and then we can be completely confounded, and that's why, you know, math is so special, because we don't just say,"Oh, that's probably right." We think about why it's right, and we try and prove it. So I will finish there with just a little plug for my next lecture, which is on the beauty of geometrical curves, where you can see this lovely thing on March the 14th. All right, thank you very much.(participants clapping)- Is the golden ratio considered to have aesthetic properties across the world, or is it just a European phenomenon?- So I've, in the writings I've seen about it, it does seem to be quite a Western idea, and as, you know, as I said, actually, not necessarily even a Western idea, because this book The Divine Proportion actually didn't really make claims for it, beyond, you know, that it's lovely properties that are due, to do with the mathematically beautiful solids, like the Platonic solids. So yes, it does have associations with beauty, but even in kind of the ancient Western thought, they didn't say it's, these are the proportions of the human face, or, you know, Vitruvian man, not about gold proportions. It's about fitting into squares and circles, and if you look at the actual ratios that are stated by Vitruvius about the human body, not one of them is the golden ratio, so, you know, maybe the answer is not even Western. (chuckles)- [Participant] Mentioning the historical origins of the measurements of A4 paper, I had occasion, I was reading a mathematical economics text, and I'd forgotten how to solve a differential equation, and I went back to my notes as undergraduate 60 years ago, and they're on foolscap paper.- Oh.- [Participant] Is there any mathematical connection between the dimensions of foolscap paper in much the same way as there is to A4 paper?- I don't think, someone may correct me, I don't think foolscap paper has that same property. I mean, it's about, it's just a little annoyingly bit bigger than A4, right? So if you have an A4 folder, it won't quite fit in. I don't think, and in all those wonderful, one does miss sometimes, I know, you know, I love the A4 and all of that. I love it, I love the mathematics of it, but, you know, Double Elephant, and Imperial Eagle, and all of these wonderful old names for paper sizes, you do miss them a bit, but yeah, I don't think that foolscap has that property, no.- [Participant] Thank you.- [Participant] Am I correcting saying that one mile is 1.61 kilometers? Is it actually the exact value, the gold ratio, or is the French Revolution something?(instructor laughing)- Oh, that would be, I would love it if that- It's, I believe it's a coincidence, but it's a very useful coincidence, because then you can use the Fibonacci numbers to convert miles to kilometers in your head, right? 50 miles is 80 kilometers, that's Fibonacci, right? So it's a very useful thing to be aware of, and occasionally, I remember a time when it feels like it's impossible, but I believe there was a time when the exchange rate from the pound to the dollar was also one, (chuckles) like 1.6. Those days feel rather distant now, but then again, it's very easy to convert backwards and forwards, because of this one over phi equals phi minus one thing, so instead of having to do one over something, you can just take away one, that's really, really useful. So I don't think it's on purpose, but it's certainly a nice property for kilometers and miles to have.- Last quick question, which is an online one, it's a Fibonacci one, you'll be pleased to know. Given that the Fibonacci sequence is divergent, and its terms tend to infinity, does it have a limit? And if not, is there a set range of values to consider as significant?- So it itself, the questioner is quite right, that the terms just get bigger and bigger and bigger, so it does, it diverges. It goes off to infinity, but the thing we can say is when we take, you know, these ratios, they are approaching a limit. So you know, it, as you get further and further up, each term will be about 1.61833, whatever, times the previous one, but yeah, the terms themselves do go off to infinity.- Professor Hart, thank you very much for another tremendous lecture.(participants clapping)