In Our Time - The Fibonacci Sequence

Episode Date: November 29, 2007

Melvyn Bragg and guests discuss the Fibonacci Sequence. Named after a 13th century Italian Mathematician, Leonardo of Pisa who was known as Fibonacci, each number in the sequence is created by adding the previous two together. It starts 1 1 2 3 5 8 13 21 and goes on forever. It may sound like a piece of mathematical arcania but in the 19th century it began to crop up time and again among the structures of the natural world, from the spirals on a pinecone to the petals on a sunflower.The Fibonacci sequence is also the mathematical first cousin of the Golden Ratio – a number that has haunted human culture for thousands of years. For some, the Golden ratio is the essence of beauty found in the proportions of the Parthenon and the paintings of Leonardo Da Vinci. With Marcus du Sautoy, Professor of Mathematics at the University of Oxford; Jackie Stedall, Junior Research Fellow in History of Mathematics at Queen’s College, Oxford; Ron Knott, Visiting Fellow in the Department of Mathematics at the University of Surrey

This BBC podcast is supported by ads outside the UK. What makes people want to believe in aliens? I'm Tristan Redmond, one of the hosts of the Global Story podcast from the BBC. Donald Trump last week announced that he'd be releasing the US government's UFO files. So why the renewed interest in life out there? And what deeper spiritual meaning might people be searching for? Check out the global story. We are serious journalists on BBC.com.

or wherever you get your pods. Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.com.uk forward slash Radio 4. I hope you enjoy the program. Hello. 0-1-1-2-3-5-8, 13, 21, 34. I could go on ad infinitum. This is the beginning of the Fibonacci sequence,

a string of numbers named after, but probably not invented by the 13th century. Italian mathematician Fibonacci. It may seem like a piece of mathematical arcania, but the Fibonacci sequence is found to appear time and time again among the structures of the natural world and even in the products of human culture. From a parthenon to pine cones, from the petals on a sunflower to the paintings of Leonardo da Vinci,

the Fibonacci sequence appears to be written into the world around us. But what does it signify? With me to discuss the Fibonacci sequence, a Jackie Studdl, junior research fellow in the history of mathematics at the Queen's College, Oxford, Ron not visiting fellow in the Department of Mathematics at the University of Surrey and Marcus Giussotoy, Professor of Mathematics of Wharton College, Oxford. Marcus Josotoy, the Fibonacci sequence named after a 13th century telemetian, a mathematician.

He was Leonardo of Pisa at the time. Can you tell us a little about him? Yeah, he's an incredibly important person, really, in the history of mathematics, because Europe before Fibonacci really was in the dark ages mathematically, definitely. I mean, they were still using Roman numeric. They hadn't got the developments that have been happening in the East. India had developed these wonderful numerals, naught one, two, three, up to nine.

And it's Fibonacci who actually learned about this and brought a lot of these ideas to Europe. So he's a key sort of linchpin in the whole history of mathematics. And he learned it actually by traveling in northern Africa with his father, who was collecting taxes and a merchant for the state of Pisa. And so it was there that he actually, became aware of the exciting developments that have been happening in the East mathematically. And you mentioned actually in your opening that maybe the numbers shouldn't be credited to Fibonacci.

And actually you can find in India the discovery of these numbers much earlier, say in the 6th century, related to rhythms. I mean, actually mathematics and music were very connected in India. And these numbers actually can be used to count how many rhythms there are where you have, if you've got long and short beats, so for example in poetry, the number, if you've got four bars, four beats in a bar, how many different rhythms can you fit in with long beats and short beats? Well, you could have two long beats, four short beats,

or long, short, short, short, short, short, short, short, short, short, short, short, short, short, short, short, that's five, and that's a Fibonacci number. And you find that actually these numbers were used and discovered in India to be able to count the number of rhythms. So it may be that Fibonacci had also been aware of these numbers through his travelling in northern Africa. But he is really seen as the beginning, I think, of mathematics in Europe because he brought these ideas to Italy from the northern Africa. We've talked about numbers developing in India and then going into the Arab world.

What about the Fibonacci sequence itself? Can you tell us a bit more about where that might have come from? Yes, yeah. Well, I mean, he describes it in a book to do with a problem about rabbits, which I'm sure we'll come on to, but the idea is that this is a natural sequence about growth. And I think, actually, this idea of Indian rhythms is a nice place to look at it, because if you want to know how many rhythms there are with a certain number of beats in the bar, with these long and short things, well, it's built out of the numbers that you've built on smaller

bar, smaller, when you have, for example, if I want to count how many rhythms there are with five beats in the bar. That's built out of the ones with four beats and three beats because you add either a short beat on the end of that or a long beat. So the Fibonacci sequence is got by adding the two previous numbers together. I guess we haven't actually said that. That's really important. The Fibonacci sequence, how do you build it up? Well, one, one, two, three, five, eight, 13, you get the next number in the sequence by adding the two previous numbers together. So it's a sequence which is very much to do with growth. The information you've got beforehand, the two numbers before the number you're trying to work out are used to get the next number in the sequence.

You add those two together to get the next number. And that's why we see it in so many different problems, because it is about growth, learning from what you've done before to get the next piece of information. Jack Estudell, he wrote his knowledge into a great book called the Liber Abaki, the Book of Calculation. It's published in 1202. Can you just give us some overview of what was in that book? Because the Fibon Archieck was only part of it. but he was bringing vast tracts of new mathematics to Europe?

Yes, he was. Trances, I should have said, not tracts. Yes, I don't quite agree with Marcus that Europe was in the dark ages before that. Islamic Spain had a very rich mathematical and scientific culture for two centuries before Fibonacci brought this work to Europe. But his book was important in spreading these ideas and in writing them as a textbook, really. It's the first European textbook in mathematics.

It's an enormous work. The modern English translation has 600 pages, so imagine that written by hand in manuscript. It's a very large work. The beginning of it is introducing the Hindu-Arabic numerals. The very first line of it introduces those numerals and explains to you how to write them and how to calculate with them because you need to know not just how to write them. 1-2-345. Well, he writes them 9-8, 4, 6, 7. He writes them backwards in the Arabic way. But he has to explain to you how you add, how you subtract, how you might. multiply using place value, which comes with the numerals. Can you just explain place value a bit more? Yes. If I write the number 666, 666, the first six stands for hundreds, the next six stands for tens, and the final one is units. So according to the place that the digit takes, that tells you something about the size it's representing. And it means you only have to learn nine digits plus zero, and then you can do all the arithmetic. you want to do. So it's a very powerful

system. So he started with that. What else did you develop inside this great book? Well, after seven or eight chapters on teaching you how to use these numerals, he then goes on to lots of practical calculations. There are lots of problems in commercial arithmetic, for instance.

For instance, if you buy so many yards of cloth at such a price, how much will it cost you to buy so many other yards? That kind of problem. Also, converting currencies, because at that time, town had its own currency more or less so you needed to know how to convert

currencies between different places. Lots of problems of that kind. And then there's a long chapter on recreational problems and that's where the rabbit problem comes. There's more. I mean after that there's a very short chapter on algebra and there's a very short chapter on geometry. So it's really a compendium

of the known mathematics of the time that he'd gathered from his travels in North Africa, Middle East. You pointed out that some of the This has been going on in Islamic Spain for a couple of hundred years. But would you be true to say that Fibonacci brought it to Europe in a more general and a more influential manner? He brought it to Italy. I mean, Spain was enormously influential.

And lots of material that eventually found its way to Northern Europe actually came through Spain, not through Italy. Fibinacci's book had a relatively limited circulation. But it was important because it certainly spread the numerals through Italy. eventually into France and Germany. Ron Nott in chapters 12 or 13, or so I've been told, I have not read this book. Fibonacci sets out various problems, which are in need of mathematical solutions.

One problem he talks about it, is the Fibonacci sequence, and he uses rabbits as an illustration. Can you bring the rabbit? I'm not going to say bring the rabbit out on a hat. Yes, so it's about doing what rabbits do naturally and it's just introduced

the idea of perfect numbers and this big chapter that Jackie mentioned it's about a whole quarter of the book after the things that we now learn in primary school and it looks if it might have just been perhaps an arithmetic exercise because although it's about rabbits that you put a pair of rabbits in a field

and they can reproduce after one month and each month after two months and each month they then produce a pair and so he says if I've put a pair in an enclosed field, how many will there be in 12 months? Well, each rabbit continues to the next month,

we're assuming there are no foxes around or other diseases that we hear on the news these days. And so he says that after one month you've still got your original pair, they produce another pair. And then the next month, the original pair have produced a new pair, and the pair that were there from the month before now become reproductive. And so you get one pair, and then the original pair plus the new pair two,

then the original pair plus a new new pair, and the original one three. And it goes on that way, and at the end of... And these are the numbers, he says each month is the sum of the two before. It's the sum of the two before because all the robes that were alive the month before are still there.

And then all of those that were alive two months ago are the ones that are reproducing. So for each one of those, you get a new pair. So it's the sum of the last month and the new ones from the month before. So that's why you sum the two before. And you're starting with a pair when you start and a pair when they're a month old.

You start from one and one. And it builds up in that way. And it's just a very small problem in this huge chapter, as Jackie said, on all sorts of practical problems about ploughing fields and cloths and things. And they end up in here with 377 pairs? That's right, yes. 1, 1, 2, 3, 8, 13, 34, 55, 85, 89, 144.

And is it there, he stops all the next one, forgotten. So this is just a small bit of one chapter, this? Yes, it is. It wasn't a very significant one. It was only later on a French mathematician, Edwa Luchas, writing in the 1800s, who wrote mostly about the sequence, and in fact gave it its name. It just wasn't known as the Fibonacci sequence. It's one of many sequences like the perfect number in the chapter before,

sequence of perfect numbers. And he was the one that gave it its name, Leonardo of Pisa. He writes, he calls himself Fibonacci. We're not sure whether that means the son of the Bonacci family or the son of good fortune, but that's the name that stuck from the 1800s. Why is it when it take so long to get out and get taken up? Well, it wasn't known before. Lucas looked at the mathematics of it,

and another similar sequence, instead of starting with one-on-one, you start with two-and-one, and you get a different sequence and call the Lucas numbers two and one with the same principle, two-and-one make three, one-and-three, one-three-and-four, make seven, and those two are very much tied up mathematically, and that's the name of, well, that sequence has his name. But they were known beforehand Keplerats about them in the 1600s. He hints that the idea of patterns in the vegetable kingdom, as he calls it,

and he writes about the golden section as well, which again is another number related to this series. Maybe we'll come onto that later. But they're all very much linked together. The Fibonacci numbers, the Lucas number, and the golden section pair of numbers, really. Well, let's look at the golden ratio with Mark as you're so at this golden ratio. This golden section golden ratio, how do you get it out of the Fibonacci sequence?

And what does that signify? Yeah, so you get it, if you look at the ratios of one number to the previous number, so for example, 8 over 5, and you keep on doing it. So let's take the next one, 13 over 8. So these fractions, as you go through the sequence, converge more and more on a, special number, which we call the golden ratio. And the golden ratio has been studied since the ancient Greeks, and it expresses what we sort of believe as the perfect proportions for a rectangle. A rectangle is in the golden ratio, if the ratio of the long side to the short side is the same as the ratio of the

sum of the two sides to the long side. And it's a very sort of aesthetically pleasing sort of ratio. And I mean, one starts to find it sort of everywhere. I mean, people find it too many places, I thing. But for example, the Parthenon is believed to be constructed in this beautiful ratio. And a lot of frames in the Louvre, for example, like this sort of shape. They don't like squares. They like this rectangle. And postcards. And, yeah, so you do find it in a lot of places. And Ron's showing me his ticket from the train. That's in the golden ratio as well. So your credit cards and things like that are probably. but there's this relationship then between this very special ratio and the Fibonacci numbers because the ratio of one Fibonacci number to another as you go through the sequence

converges on this golden ratio. And it's a very important relationship between these because what mathematicians want to do is to find efficient formulas, for example, for calculating the Fibonacci numbers. If I want to calculate the millionth Fibonacci number, I don't want to have to count all the pairs all the way up to a million because that's a terribly inefficient. We're very lazy at heart mathematicians. So the golden ratio gives you a lovely formula. You take powers of the golden ratio essentially and you can use those to get the millionth Fibonacci number without having to do any work at all. So if you're actually

calculating and wanting to know what Fibonacci numbers are, the golden ratio is extremely important in capturing them. Jack Estella, what Marcus has mentioned the Greeks, the ancient Greeks, as we keep calling them, knowing about the golden ratio, what did Euclid have to? What did Euclid write about that? Well, the construction that Marcus is talking about appears in Euclid, in book six of Euclid,

where Euclid sets as a problem, how do you divide a line into two segments so that the ratio of the whole line to the longer segment is the same as the ratio of the longer segment to the shorter segment, which is exactly what Marcus was describing. And this is quite a difficult construction in Euclid, but he tells you how to do it.

He explains how to do it. And it became very well known. It was known as dividing a line in extreme and mean ratio. But all later mathematicians knew this, because they knew Euclid very well, of course. So it's there. I'm not sure that Euclid calls it the golden ratio. I don't know where the term golden ratio comes.

Plato knew about it and talked about this particular ratio. What's your view of the notion that buildings like the Parthenon followed this ratio? I don't know. as Marcus said, I think people can see this ratio too often in too many things. It does give pleasing proportions, so maybe people just built in pleasing proportions, and that happens to be close to the number we now knows the golden ratio.

Whether they actually had that in mind as they built is doubtful, I would say, but I don't know. But does it suggest that if they didn't have Euclid in mind, the fact that they did build something which does seem to follow those proportions means that it's instinct in human beings. I mean, I'm just asking. Yes. Yes. Yes, it's certainly an instinctive... So where does that come from?

I don't know. I'm not a psychologist. I think you can see... It's again about this way of growth and building things up because you can build this perfect sort of rectangle by using the Fibon Archie sequence in a way. I mean, you take a sort of a one-by-one box, and then you add on the side of that another one by one box.

So now you've got a one by two sort of building. Now what do we know about? Well, we've got the number two. So let's add a two by two box on to that. And so now I've got a two by three box. Well, I've got three now. So let's add a three by three box. So now I've got a three by five box.

And you can build up what's actually sort of a spiral of boxes. And that is actually, that natural thing is growing into a rectangle, which is getting closer and closer to the perfect, to this golden ratio. So I think it naturally... But a building like the Parthenon wouldn't have been constructed on those... Absolutely not, but I think that's why we're naturally drawn to this sort of shape because it has this growth in it,

and that's why you start to see it all over nature as well. Ronnott, Euclid, I'm told, I read, seem to relate the golden ratio to the five platonic solids. Now, can you take us into that? Yes, these appear in the Euclis elements, well and some people think it was sort of well one of the aims of Euclid's 13 books was to describe the five shapes that Plato had mentioned which are the most symmetrical 3D shapes in other words if you're trying to design a dice in other words so you want all the edges to be

the same length all the angles to be the same and the faces to be identical and so the usual shape we use is six squares to make a cube but there are five shapes the simplest one has four faces which are regular triangles four faces tetrahedron. The next one is two square-based pyramids glued together on their squares, and that makes eight triangular faces, the octahedron. There's the hexahedron, the cube. And there are two other ones. There's one with 20 faces, triangular faces call the icosahedron, and then another one with 12 pentagonal faces call the dodecahedron, and these are the five most symmetrical ones. And if you look at the mathematics of the icosahedron and the dodecahedron,

that's full of ratios to do with the golden section. In fact, if you describe the coordinates of these shapes, as we might do in a take-art system, in an ordinary 3-D coordinate system today, the tetrahedron, the cube, and the octahedron, just have plus ones and minus ones. The icosahedron and the doetachahedron, you also add on multiples of,

you also add on the golden sections, these two numbers, pluses and minuses, and that gives you the shapes. So he was thinking about that at the same time, so they were onto that at that time. Much the same, or was the Fibonacci sequence of refinement? Well, I think it's interesting with a, I mean, I think the golden ratio, you see, was around before the Fibonacci sequence ever was ever around. I mean, this, and I think it reflects, actually, the ancient Greeks were more interested in geometry and ratios of lines. And I mean, I think you can even find this golden ratio in just a simple pentagon. And the Pentagon, of course, was a very sort of spiritual figure amongst the Parathagoreans. And so it's interesting that then you get the numbers, these Fibonacci numbers, starting to mix in much later, sort of, medieval time and then there's this sort of connection between geometry and numbers, which is actually

a very late sort of idea. I mean, that's really what Descartes did was to fuse geometry and numbers. Before that, they were not really seen as two things together. So I think it's quite an exciting moment when you see the Fibonacci numbers actually giving rise to geometry. Jackie. Yes, I agree. I mean, this is the extraordinary thing about it, that the golden ratio was known for a long time. The Fibonacci sequences are quite separate development. It comes from separate routes. It comes from Arabic and Indian roots, whereas the golden ratio is coming out of Greek mathematics and tying these two things together

is very interesting. I don't know when it was first discovered that these two went together. I might just say the ratio that Marcus is talking about, if you calculate it, it's a number that begins 1.618, I think, and goes on, it's an infinite decimal. There's an exact expression for that. It's one plus the square root of five

divided by two, which the Greeks would have known. They would have known it in that form, which is a very precise measurement. also is interesting because it's an irrational number so although the Fibonacci numbers are giving you fractions they're the fractions which approximate but never quite meet the golden ratio and in fact they give the best approximations as well

to the golden section as Marcus said if you take the ratios of the numbers the larger over the smaller you'll get it tends very quickly to 1.618 blah blah blah but if you take the smaller to the larger you'll get 0.618 and the interesting thing is it's exactly one larger and that's what defines

these two numbers very precise and in fact makes them unique. The Euclid, we'll stay with Eucl from Marcus, Marcus Sotoy. Mathematician Luca Pacioli published on divine proportion in 509, as I'm told, illustrated by his friend, we think Leonardo da Vinci, and that we have the golden ratio in that. Can you tell us about how he went over, if you believe he did, into painting? Well, yeah, there are certainly sketches that Leonardo made,

which Leonardo da Vinci, rather Leonardo DePisa, which seemed to indicate that he was setting up proportions, for example, in faces. That he, and this is another place where you find, if you want to, the golden ratio, is in proportions of the structure of the face. So you can see that there was, I mean, Leonardo da Vinci had a lot of mathematical background, so he was, he drew a lot of these platonic solids, for example, as an exercise in perspective. I think, again, what you find is a square, if you have a square, your eye is naturally drawn just at the centre of the square.

And that's kind of uninteresting, actually. But if you have a rectangle in this golden ratio, there are other focal points, partly due to these sort of Fibonacci spiral you get with these sort of squares that I build up the sort of shape, there are natural places in a picture that your eye is drawn to, which aren't the most obvious places you might think. And so if you look in pictures of the,

done by something like Leonardo da Vinci. You find these four focal points, interesting things happening there. Now again, you might say, well, is this just somebody who's intuitively drawn to that, or have they actually, you know, actually programmed this in

and constructed a picture which has these focal points because he knows that the golden ratio naturally draws you to these four points, so you put something interesting in those places. Would it be fair to say, Jacquistadal, that mathematics in this period was underpinning, to some extent, aesthetics?

I think it's difficult to say. As Marcus says, you don't know how deliberate these things were, whether this is an instinctive thing to do because it's pleasing, naturally pleasing, or whether there is some deliberate construction going on. Or if we took the last supper, which everybody will know, that's been called following the gruel of the gold mean. Can you tell us if you think it does

or if it could have been arrived at, naturally in inverted commas, or naturally without any inverted comment? I really can't say. I think you can't say unless you know what the painter had in mind at the time. It's impossible to know whether it was... I would suggest it. I think that they were so aware of mathematics because of the perspective

and how that worked, and that relied on a lot of mathematics. They were full of platonic solids in these paintings around the time, and so they would have known about things like there were a relationship of platonic solids to the golden ratio, that I think there, But again, I'll come off the fence and say I think they probably did, know what they were doing.

I won't come off the fence. I think I'm with Jackie as well. I think when you look at pictures, there's so many lines you can draw on. Somewhere in there, you'll find one that's more or less around 0.6.1. Now, whether that's deliberate or it's a product of psychology, maybe I'll sit on the fence.

But I think certainly, you know, in more modern times, you find a lot of artists, composers, and things deliberately using all these things. Oh, definitely, yes. I mean, certainly, you know, maybe we can't say anything about that time. But we know now that a lot of composers and architects and Salvador Dali certainly was deliberately using the golden ratio in some of his work. And CUPAC as well, yes. Yeah.

Can we, Ron Nott, can you tell us about the math, we touched on with a hung jury on the aesthetic question. But what purchase did it have on other areas of mathematics? The Fibonacci. Yes. Well, in the 1800s, the Victorian towns, there was a lot of interest in kind of unifying mathematics and finding theories for things. And there was quite a lot of interest in,

they'd notice that the Fibonacci numbers appear in nature, in plants. Plants as opposed to animals, really. Yeah. And so it was, for instance, if you cut a tomato across the middle, you'll find it's in two or three segments. But look on the outside, and there's a little green bit, which has got five, four is missing. if you look at a cauliflower

and the florets are actually in little spirals There are about five spirals in one direction and eight in the other If you look at a pineapple There are three fairly obvious sets of sparrows on the outside One of them's five spirals going around fairly shallow Eight or a little bit steeper and thirteen steeper still It's these particular numbers four is missing

It's not very often there in the world of nature So when you're looking at the food we eat every day This is a series you're eating every day in your fibre a day. It puts the fire into fibonacci, the fire into fibre. And it seems to be that it's to do with the packing problem, that if you're trying to arrange leaves around a stem,

you don't really want them to overlap and cut out the sunlight or not catch the rain. Or if you're arranging seeds on a seed head, you really want an efficient packing. And it was in the... It was only proved relatively recently in the 1990s that the best form of packing is to do with arranging seeds

with a golden section. You put 0.618 seeds per turn It's as if it's a little rotating stem. This is the original stem cell at the end of a stem and where the plant grows. And it's as if it goes 0.6.18 of a turn and produces a new stem cell, which might become a leaf or a branch or a seed or a petal.

And then it goes another 0.618 and so on. And when you plot that out, quite magically, the human eye picks up spirals. Near the middle there are spirles in two directions, threes and fives, 5s, 5,000, 8, 18s and 13s. And then the mathematicians have proved that that's the best efficient, the most efficient packing, no matter how large you go.

There are no sort of seeds squashed in one direction with spaces and another. So mathematically it fits into nature and there was a lot of interest in that before it was proved around the 1900s as well. Jackie, do you want to take that up and speculate as to why you think... I know, no commitment is required and that speculates as to why you think that this recurs,

this sequence. The sequence. Well, I think Ron's explained it very well. well that mathematically certainly in seed patterns it gives you the best packing of the seeds and the most I don't know the most efficient way

for the plant to grow and to reproduce it. Yeah I think it relates to this way of building up this spiral of squares. I mean if you think a cell starts off with just one cell and then it has to add something so it adds this other cell and you get the 1-1 and then you add it's sort of grown and it needs to enlarge and actually this is why you see the

Fibonacci numbers coming up in the way that snails grow. I mean, the spiral of a snail actually has the Fibonacci numbers embedded in them as well. And if you take these squares and you build them up and draw a sort of spiral through it, you see a sort of natural shell appearing. I mean, the other one

that we have to mention is, of course, petals on flowers. I mean, that that again seems to always be Fibonacci numbers or a double a Fibonacci number. I mean, it's a very rare plants which actually don't have Fibonacci numbers. And if you do go out to the garden and

test it in the summer and it hasn't got a Fibonacci number, I bet you it's just because the petals fallen off the flower, which is how mathematicians get around exceptions. Yes, it is. It certainly is prevalent there in nature, but actually in the plant kingdom, not so much in animals. With animals, we've got this bilateral

symmetry, the mirror symmetry, which is actually very rare in flowers, and that seems to be the distinguishing, and we can't find really the golden section number very much in animals. And it's usually the series, but then again you look at a fuchsia, which has got four petals, or a defatil which has got six and those aren't the Fibonacci.

The explanation for that I've heard anyway is that it's a double flower basically that it's a three on a three and a two on a two. We've been fine anyway to get around these things. That can be if you're an ardent, Fibonacci, yes. Well, somebody did send me a picture of a flower which had seven petals on and that I can't. I can't get around. When you say it's not in the animal kingdom, of course, was the very famous drawing by Leonardo of the man in the pentagon.

That was the Vitruvian. His picture of Vitruvius's description. But Vitruvius' description is very much. This is the one you see on the front of pads of paper, the man in the square and the circle. But it's very much to do with fractions. He got a little scale at the bottom to show that it was designed with fractions. The whole thing about the Fibonacci numbers and the golden section is that it's an irauchial number. These fractions are the best ways to get at it, but it's actually not a fraction. You've described the reason for it in terms of packing.

These numbers pack well, as it were. Is there any other reasons being put forward why that numbering sequence should be so widespread? Or just, I mean, you put forward a very utilitarian one. You know, we would like a little mystery and magic. Well, in fact, if you look at cactus and succulents, it's quite often to find the arrangements of the spines is sort of fours and sevens, sevens and elevens.

And in fact, that gives us another sequence, starting with 1 and 3, 4, 7, the same principle, add the latest two to get the next 11. And that gives us these Lucas numbers, those appear. But the thing is, no matter what two numbers you start with, start with any pair of numbers, and keep adding those to get the next, and adding the latest two,

all of them get to the golden section. And it seems to be the mystery as the golden section is behind this, which manifests simply in the Fibonacci numbers, but also the Lucas numbers. Well, I think it's an important point, because mathematically it is a very important idea, this is called a recursive sequence because you use the previous numbers

to build the next one in the sequence. So the Fibonacci is the most simple version of that. You use two previous, add them together to get the next one. But in mathematics, certainly in the 20th century, you've been looking to find more complicated versions of this in many sequences of numbers. I mean, my own work, I deal with trying to count numbers of symmetries. But what I'm trying to do is to show

that there might be a similar process to build my sequence of numbers. They're very nice sequences because you get, get very explicit formulas for them. And so Fibonacci just kicked off, or this sequence kicks off, a whole way of looking at number sequences, which we're trying to find Fibonacci in as many sequences as we can, or that style, recursive functions.

So it is a very important mathematical idea. Jackie? Well, there are many very beautiful properties within the sequence itself as well, apart from relation to plants. I mean, just looking at the numbers themselves, for instance, if you take the consecutive numbers, two, three, five, If you square three, you get nine.

If you multiply two and five, you get ten, which is one more. This happens all the way along the sequence. If you square any of the numbers, it's one more or one less than the product of the numbers either side. And there are many, many, many such properties. It's just amazing. It's absolutely fantastic. Which is why there's a whole journal, the Fibonacci Quarterly, which is dedicated to discovering all these weird and wonderful things.

Could we have some more, please? Ron's the expert. Well, that's one. The other one is if you start from one and one, your indexing sequence are the first sighting. Fibonacci number is one, the second is one, the third is two, the fourth is three, and so on. Using that number sequence, then since three divides into six as just numbers, then the third Fibonacci number two divides into the sixth, which is eight. And this happens whenever, if you start with that number system,

and so in fact the Fibonacci numbers in the prime positions are the ones that become important. And in fact, this relates to Marcus's work, because each of these is distinguished by a very unique prime, which is itself a divisor of this Fibonacci number, it acts as a kind of marker all the way through, and then it relates to... Of course, but there's an open problem, which is, are there infinitely many Fibonacci numbers,

which are prime numbers, indivisible numbers. We don't know that. 13, for example, is a Fibonacci number, and it's also an indivisible prime number. But that's an open problem about Fibonacci numbers. What does that mean? Are there infinitely many Fibonacci numbers?

No, but what does open problem mean? Open problem means that we don't know. Oh, yeah, it means that... I see what you mean, yeah. Far more elegant, I'm going to use that. That's an open problem. Rather than I haven't a clue. So I think it probably is put you know.

I mean, if you can calculate the Fibonacci at infinitum, why don't you know? Very good, because I'm a finite being. And therefore, I mean, it's a very fair point. You see, the mathematicians are finite beings at heart. And to be able to analyze an infinite sequence like this, we have to find clever ways to show why infinitely often a prime will pop up. And this is a mystery. The other interesting place I think where Fibonacci numbers come up

is actually in generating Pythagorean triples. So Pythagoras' theorem is about right-angled triangles. So you can build a right-angled triangle with a side which has short-side three, next side four, longest-side five units. Three-four-five is called a Pythagorean triangle. But if you want other examples of triangles, right-angle triangles, each of whose sides are integer lengths, actually it turns out that the Fibonacci numbers

are a wonderful way of generating these triangles. So it shows these Fibonacci numbers are just embedded everywhere. They're embedded in geometry, they're embedded in nature, and also aesthetic things like music architecture. It comes back to that. If it isn't, the problem isn't the record, but it's a fairly accurate paraphrase. Galileo said, I discovered the book of the universe

that's written in the language of mathematics. That's what you seem to be saying now. Yes, I think so. I think that nature, is built by very simple operations, which can produce things with immense complexity. And Fibonacci sequence is a very simple way of generating growth, and yet suddenly you have just a very rich sequence.

Ron, you mentioned earlier LeCobusier, and you said, Jackie, firmly fence-bound, would not come up the fence as well that they'd known about the government when they did the Parson or not, whether Leonardo really knew about it. But when we come to the modern age, when it's, then people are taking it on. Can you tell us how commutia take it on and incorporating it and making central to their work, self-consciously?

Yes, it was, again, it's the early 1900s, and he wanted to design buildings which had a human feel of human proportions. And so he had the sort of model of the human, which is that if you look at the position of your belly button, your navel, it's about this golden section point further up. And if you raise your hand, well, if you look at your arm, the length of your hand or the length of the... up to the elbow, looks if it's a better of golden ratio,

and the elbow to the arm. And so he kind of formalised that in a structure called the modular. And he used that to sort of tell his architects to design buildings. On the other hand, he did say, but if it's not working, if it's not really fitting, then abandon it. But it certainly, he was trying to use it as a guiding principle in designing buildings.

And once the, as it were, the Piminacci sequence K came out, which was centuries after it had been, It was in this original book. It was taken up or developed. There's a Scottish biologist called D.W. Thompson on growth and form in 1917. How did he think come in on this argument? Yes, he wrote a book on growth and form. Again, looking at the 1900s when they were trying to find principles behind things,

major principles. Marcus can tell us probably about the Hilbert program later on for mathematics. And that led to a whole growth of mathematics in the 1900s. He was writing about patterns of can you take the shape of a horse's head and relate to a sheep's head. They're similar. There's a small defamation. And he wasn't the first to write about this.

It relates to a book really by a church. He was the first person in about 1910 to formalise shapes of objects in nature. And then there was a big chapter in there about the golden section and the Fibonacci numbers. They didn't know quite why they appeared, but they were noting they were. and the properties. And there's one other interesting property we haven't mentioned about the golden section

which I call Fai which usually call Fydeus the architect from who designed the sculpture in the Parthenon. And that if you take the powers of the Fibonacci, you can add the two powers,

the squared plus the cube power adds to give you the fourth power. The third and the fourth power add to give you the fifth. Well, you'd expect a multiplication but you can do it with an addition. And again, that's a very distinctive property of these golden section.

numbers and purposes why it's there in nature. It can do multiplication by addition. Do you see, is it too fanciful? I'm certain it is, Jackie. I'm looking away as I'm speaking you. The fact that it appears everywhere, it appears in many, many places. Does it suggest some sort of a platonic attribute? There's, as he would say, a divine form behind all this? That is too fanciful for me. I can't go there. It's... It's not the sort of thing I can speculate about. Can I turn the question or the conversation around? Because I'd actually like to ask a question of these two experts,

since we have them here, which was when the Fibonacci sequence was rediscovered, as it were, because Fibonacci's book wasn't, the text wasn't published until the 19th century in Italy. Until then, there would have been early manuscript copies, but no printed edition. So where did people learn about these numbers, or were they rediscovered independently by Lucas and others? The numbers themselves It's just a very small problem The main thing is as you said

The book is about arithmetic That we learn in primary schools now And the numbers And that certainly went across Europe and everywhere So it was quite influential Well I guess Lucas probably is But where did Lucas find it Or did he rediscover them for himself

That I'm not sure It's a French book I've tried to I've tried to find a copy of the book And I can't It's quite difficult Well again Lucas would have been I mean these Lucas numbers

Which are built in a similar way Of course very important For testing Mersenne Primes And that's how he So there are lots of the point is because they crop up so many places, there are loads of... I think the other interesting plays we must mention is music. I mean, architecture, it's sort of...

You can see that, you know, the way things are being built, boxes being put together. You can see the Fibonacci sequence. But I think the interesting thing is people like Bartok and Debussy who are threading it, and I think, again, deliberately, you see these numbers in the sort of bars that they're building up the piece of music. And again, it's this idea of starting with something simple,

which a piece of music generally does, and developing themes, drawn to the Fibonacci sequence as a structure within which to write. I work do some work with a composer called Dorothy Carr from New Zealand. And she has a piece which just has the Fibonacci sequence as its skeleton, the cello piece. And she loves it because she sees the way it builds on the two things which have gone before, build to create the next piece. And so I think, actually, a student of mine yesterday told me even a band in America called Tool,

use the Fibonacci sequence. So, you know, in music it is a nice structure. again because there's aesthetic sort of growth and climax in it, I think. Do you think, Jackie Sell, that more discoveries are going to be made out about and through the Pibonacci sequence, that it's going to yield entrances to other areas? Undoubtedly. I don't think we've, we certainly haven't come to the end of all that can be known about it or through it. And as Marcus said, there are open problems. There are still problems there. Yeah, I mean, I'm trying to prove that my numbers are basically like Fibonacci numbers,

but I can't do it. Your numbers being what? Numbers which counts groups, symmetries, something completely unrelated. But some of the theorems I have proved are showing that the structure of these numbers are very similar to the ones that Fibonacci do. So, you know, for a modern mathematician,

this style of mathematics is really important. Yes, the Fibonacci Association in America has a journal that comes out, but every two years they have an international conference. I've been to them for a few years now, and it's quite interesting to see still the development of new ideas in here. And I was talking to a firm in California

who are using the shapes of the seashells with the Fipanacci, with the golden section spirals. They've taken the centre of one of these and made it into a rotor. And it tends out to be about 30% more efficient than any known rotor. They've been painted in a few years. And just this very small rotary

and I saw in a huge tank just sort of rotates a tank. It's designs based on nature, which seems to have new applications. And they've even found their way into popular culture, of course, because anyone has read Dan Brown's Da Vinci Code, the first code you have to crack is the Fibonacci numbers. So, you know, I think they're really in the public imagination.

So, you know, what better advertisement do you want than, you know, how important these numbers are? I don't think they want to. And there's now an English translation at last of the Libra Abbeki, so anybody can read this for themselves and all the other wonderful things that are in the book. Well, thank you all very much. Thank you for guiding me through that.

No, I was less than it. I just hope I can remember it. Next week... It's very simple, you just add the two previous ones. I can remember that, but all the other stuff you've said. Anyway, thank you very much to Jackis, Sirle, Marcus, to Sotoy and Ron. Not next week we're talking about mutation,

in genetics and in evolution. Thanks for listening. We hope you've enjoyed this Radio 4 podcast. You can find hundreds of other programmes about history, science and philosophy at BBC.com.uk forward slash radio 4.