So I. When I was at school, I was very frustrated by the choices we were kind of asked to make. It was a Shakespeare or the second law of thermodynamics Rubens or Relativity. WC or DNA art or science. Now, at school, I found this demand to try and make a choice between these two deeply frustrating i. When I went up to secondary school, I actually went to school, a comprehensive school in Oxfordshire, just local to here. I started learning the trumpet, started playing in orchestras.
In fact, Peter Norman, who's here in the audience, rescued me once when I was stranded here in an orchestra and there was no lift home driving all the way back to Henley on Thames. So and I did a lot of theatre at school and but at the same time as I started learning the trumpet, it was about the time that I fell in love with the world of science and in particular mathematics. The wonderful power that mathematics has to tell us where we've come from and where we're going to go next.
And I seem to have to make a choice between these two. And I was better at my mathematics than I was at my scales on the trumpet. And so I chose the the mathematical route. But actually one of the things I've always enjoyed about Oxford is when I came up here as an undergraduate, I found that actually I could still do both. In fact, I think the college system almost encourages you to spend your time with people doing other disciplines.
So I spend a lot of time trying to justify why I thought mathematics was as exciting as Derrida is deconstruction or various other things. And so I kept on doing my music hair and my theatre and but I went to mathematical roots. I became a professor here. But I have always kept my interest in the arts. And as time went on, I began to realise more and more that this idea of the two cultures that C.P. Snow talks about, it's really a false dichotomy.
And in fact, the more and more time I spend studying, spending time in the arts, working with all creative artists, I realise again and again how often we're both drawn to very similar sort of structures. So what I want to do in this presentation is to I've chosen five of my favourite 20th century artists from various different disciplines, and so I've called the talk the secret mathematicians, because in some sense I think that these artists are a bit like secret mathematicians.
They're being drawn sometimes consciously but often unconsciously, to structures that are a fundamental importance to me as a mathematician. So the first artistic discipline I've chosen is one which has a lot of resonance with mathematics traditionally, which is music. So I've chosen a composer. One of my favourite composers from the 20th century is Olivier Messiaen, who is obsessed with mathematical structures as an inspiration for a lot of his work.
And in particular, there's one piece that he wrote while he was a prisoner of war in the Second World War, he was in a prisoner of war camp and there was a rickety upright piano. There was a clarinettist, a violinist and a cellist. And he wrote this piece called The Quartette. For the end of time. The quartette is meant to kind of represent that that incredibly desperate period of history.
And the first piece, the liturgy, the crystal, is meant to capture this sense of unease and a sense of never ending time. And the way that Messiaen so sort of captures this unease is actually to use a little bit of mathematics. In fact, he uses two numbers that are very important to my own research prime numbers, so indivisible numbers. So the prime numbers he uses are 17 and 29. Now, the way that he uses this is in the piano part.
Now, when a piece opens, you hear the clarinet and the violin exchanging bird themes. Messiaen was very interested in bird themes, used to notate them as an inspiration as well. But it's the piano piece where the mathematical structure can be heard. What it is. It's a very repetitive piano piece. The rhythm sequence is just 17 notes, a rhythm repeated over and over again. But the harmonic sequence is 29 chords, which themselves are repeated over and over again.
So you have the 29 chords and then you repeat them again and repeat them again. But the choice of the 17 notes for the rhythm and the 29 for the Homeric sequence means that you get a very strange effect because the choice of 17 and 29 mean that these numbers don't come in sync until you've heard 17 times 29 chords. So you get the sense of you can hear something repeating, but then it's sort of when it's the rhythm starts, the harmonic sequence is still working its way through.
When the harmonic sequence starts, the rhythm is halfway through its pace. So if you see a score for the piano piece, so the 17 note rhythm sequence, you starts with crotchet, crotchet, crotchet, and then you get a syncopated rhythm until you get crotchet and then minim. And then you see the crotchet, crotchet, crotchet repeated again and off it goes.
The rhythm is exactly the same. 17 Notes. The harmonic sequence goes all the way up to this point where you see the same chords repeating, but you see when the records are repeated again, it's a completely different rhythm, rhythm structure. So it's very hard to hear what Messina is doing, but you certainly get a sense of there being structure there, but a huge unease as well. So let's hear the prime 17 at 29 at work in the quartette for the end of time.
So that's the rhythm sequence repeating again. The harmonic sequence is still working its way through the 29 chords. And now the harmonic sequence starts repeating, but the rhythm sequence is doing something completely different. And so the choice of 17 and 29, these two primes are chosen very deliberately by Messina in order to create this sense of unease and the things keeping out of sync.
Now. I think what's interesting is that very often at the core of this, the fact that artists and mathematicians and scientists seem to be interested in similar structures is that in some sense we're all reacting to the natural world around us and trying to find patterns in the natural world. So you actually find this effect at the heart of the survival of a very strange insect that you can find in North America. It's a cicada which has an incredibly strange lifecycle.
There was a very big brood this summer which appeared on the East Coast. It was, but they hadn't been seen for 17 years because what these cicadas do is that there are the eggs in the ground and then the the eggs produce the cicadas. They emerge into the forest, they eat the leaves, they mate, they lay eggs. So actually, this is the sound of one of these cicadas. The sound is so loud that residents in general move out for the 17th year because it's totally unbearable.
People make sure they don't plan their weddings while this is happening because you can't hear anything. But after six weeks of passing, the cicadas all die and the forest goes quiet again for another 17 years before the next brood next generation emerges from the ground. That's I had a chance to go and see these the year before last in the it was it was Nashville, Tennessee. They had a brood there. But this wasn't a 17 year cycle. This was actually a 13 year cycle.
So it's very curious. There are different species across America, but they only have 13 or 17 unicycles. There are none with 12, 14, 15, 16 or 18. So two prime numbers, 13 and 17. So it's very curious, what is it about the primes which seems to be helping these cicadas in some way?
Well, we're not too sure, but one theory is that it's very much the same principle as Mason was using to keep the rhythm and the harmony out of sync in the liturgy, the crystal, because there is a belief that there might have been a predator around in the forest, which also used to appear periodically, and the predator would try and time its arrival to coincide with all of this card is appearing.
Now the cicadas, which had a non-prime number, life cycle, they got wiped out because they got in sync too quickly with the predator. So, for example, I've got a little example here where the Predator appears every six years and the cicada appears every nine years. So they have a common factor three. So that means that the predator and the cicada will meet each other every 18th year. So every second time this cicadas emerge, they meet this predator.
And so they're quickly wiped out. But those cicadas, which have a prime life cycle. So let's shift. So let's make the cicada have a seven year life cycle. So seven means actually it's appearing more often in the forest. So you might think, well, this is got even more of a chance of getting wiped out. But no, because now six and seven are co prime. Seven is a prime number. It means that this cicadas keeps out of sync of this predator.
So the first time they actually meet is a six time seventh year, the 42nd year. And so this prime number of years gets a better chance of keeping out of sync and so surviving. So it seems like there was a quite a competition in some of the forests in North America. The Predator would move its lifecycle, get in sync, but it's the 17 year lifecycle. So if you know your primes, you kind of survive in this world. So it's a good message. There's an impact for you. You know, you, Max, you survive.
So so the cicadas knew they survived and they seems like the predator got wiped out. It didn't it couldn't get into sync with the 17 year lifecycle or a 13 year lifecycle. But it's kind of curious. This is exactly the same trick as Messina's using that the Predator, that's a bit like the rhythm sequence and the the cicada, a bit like the harmonic sequence. And keeping those things out of sync gives a sense of kind of unease and never ending time that well, almost never ending time.
The message I wanted to know. For me, what's intriguing is that he is missing who's read up on a bit of mathematics and realise that it can be useful for him. But often artists are being drawn to mathematical structures for without knowing it. And often you can find examples where the mathematics was discovered first by artists and only subsequently realised were important by mathematicians. And one very interesting example of this, it's a very famous sequence.
So if you're not a mathematician, what's the next number in this sequence? You always say, if you haven't read The Da Vinci Code, you're not allowed to have read the definition. But what's the next number in this sequence? 34 It's a very famous sequence that a lot of kids get exposure at school. It's a lovely sequence because you get the next one by adding the two previous numbers. In the sequence they call the Fibonacci numbers. Fibonacci discovered them because he realised they were very much.
Relate it to two things in the natural world, again, very much associated with growth. So the way that things seem to grow, I mean, the numbers themselves have a natural sense of growth in them. And that seems to be also apparent in the natural world. So, for example, if you count the number of petals on a flower, invariably it's a number in the Fibonacci sequence. Sometimes you get two copies of the flower, so you can get double the number in the Fibonacci sequence.
And if it isn't the number in the Fibonacci sequence, that's because the petals falling off your flower, which is how mathematicians get round exceptions. But actually Fibonacci wrote about these in relation to another problem, which is if you look at two generations of rabbits from one season to the next, how many rabbits do you expect with a particular mathematical rule? So the mathematical rule was you start with one pair of rabbits.
They take a month to mature, at which point that they can give birth to another pair of rabbits who then themselves take a month to mature before they can give birth to another generation. And so it's quite a sort of complex problem to get your head around, as you know. Oh, these ones are all mature, yet these ones are still giving birth. These are very mathematical rabbits which never die, of course, and which always give birth to a man and a female.
But Fibonacci realised that these numbers were described by the female. We apply this sequence where you add the two previous generations together to get the next one. So they've been very much attributed to Fibonacci.
But the intriguing thing is, and I only discovered this quite recently when I was looking into the subject of Indian mathematics, I did this programme about the history of mathematics and sort of discovered I mean, we teach our subject very historically and somehow you don't realise where a lot of these things come from. But actually I discovered these Fibonacci numbers that Fibonacci wasn't the first to discover it.
In fact, they were discovered already in India by poets and musicians who were interested in what sort of rhythms that you can generate. We use long and short beats, so if you allowed long and short beats, I suppose you've got four beats in the bar. So for short beats and how many different rhythms can you make? So what you can do for sure beats, or you could do a short, short, long or short, long, short or long, short, short or long, long.
So you've got five different rhythms that you can make out of these long and short beats, which was of interest to know what sort of different possibilities, where have you doing poetry or creating music? And so you say, okay, if I had an extra short piece, I've got five short beats in the bar. How many different rhythms can I make? Well, they discovered it's the same rule is for the Fibonacci numbers, because you add the two previous numbers together.
So there are eight different rhythms and actually you can see why I think it's much more obvious in there with these rhythms. And then four for the rabbits in a way, because how do you get the rhythms? So you can take all the ones with four beats in the bar and add a short beat to those, and that gives you the sum of them. But you can also take the ones with three beats in the bar and add a long beat to those. They're all different and they give you all the different possibilities.
So it seems much more obvious why these numbers, these Fibonacci numbers, actually count rhythms rather than sort of rabbits. So actually it was the musicians and poets who discovered these numbers. First they had written in Hamish. Sandra writes about these numbers before Fibonacci, and it seems that even before Ramachandra but you know, they really deserve to be called the Sandro Fibonacci numbers.
And I think, you know, I talked a lot about rhythm and mathematics, and I think that, you know, there's a kind of obvious connection between counting and and mathematics, mathematical trumpeter. I spent a lot of time in my orchestra sort of counting balls, rest, you know, seem to be all counting. 90 423, 4952, three, four. Now, I think loudness kind of summed up that sort of connection between the rhythms of music and count and counting.
He said to music is a pleasure, the human mind experiences from counting without being aware that it is counting. But as you know, I think this connection between art and music goes a lot deeper. The sort of structures that a musician embeds inside that musical composition uses a lot of sort of mathematical ideas. After all, in some sense, what distinguishes music from noise? You're looking for structures and patterns and things being related to something you just heard.
So and Stravinsky wrote very nicely about mathematics being so important to a composer. He writes, The musician should find in mathematics. His study is useful to him as a learning of another languages to a poet. Mathematics swims seductively just below the surface. And if you take a piece like Box Goldberg Variations, you can see Bach just playing around with combinatorial possibilities for the different variations.
Each of the variations is, in some sense, using ideas of symmetry to create something which is related but different to what you've just heard. If you move into the 20th century, you have lots of examples. Schoenberg, for example, he said, okay, well, we're going to throw away. Atonal music. But if you throw away structure, you need new structure to be put in place. And so he introduced this idea of the 12 tone rose and looked at it very mathematically.
He would take permutations of the 12 notes of the chromatic scale and then apply symmetrical rules to that used to create a palette of a 4812 tone rose which he would use to compose things with a and used to love this as well. And actually in one of his pieces, he seemed to discover a very significant mathematical structure by using these 12 tone rose. Purely from for aesthetic reasons, but from a mathematical perspective is extremely significant in my own subject.
One of the I of it I had this kind of dilemma whether to choose Messiaen or another of my favourite 20th century musicians, which is Xenakis. And Xenakis is extraordinarily mathematically literate. In fact, this is a piece Novus Alpha, which is written for solo cello. He dedicated to every scholar amongst us, as well as two other mathematicians who was in fact the inventor of the language that I talk every day when I'm doing the subject of symmetry.
And in fact, there's a symmetrical object hiding behind the composition of this piece of music. Actually, the score I've got here isn't for Name Alpha. It's actually for another piece called Metastasis. But you can see that the natural this score looks like a piece of geometry, not a piece of music. But I'm going to play you this an extract from Nomos Alpha for solo cello. And I want to say it's actually based on a on a three dimensional symmetrical object.
And I want you to try and here what symmetrical object is sum it up in your mind is only by the following piece. So did anyone get a symmetrical shape preparing in their mind? A circle. Interesting. And a spiral. Well, in fact, that was a cube hiding behind there. But I must say, even when I knew that, I find it quite difficult to hear the cube.
But actually it's a little bit unfair because I haven't really played you enough of the piece to really get you a sense of the cube, because as an artist does, is you want to hear what's an arcane star is is to put. He uses a cube and he puts certain musical ideas that the cello can play. So you heard the kind of pizzicato there's an effect where you can turn the bow over and hit the strings with the wooden side of the bow necklace. Sandy So these are put on the eight corners of the cube.
And then in each new variation, what's in our co-stars is a symmetry of the cube, which then tells him the new order in which these should be done. There's another cube which is controlling the sort of time spent on each of these items and then using those constraints. He allows his creativity to take over and then he plays within those particular constraints. So it's interesting because he he uses the symmetries of the cube there.
There are eight factorial different ways that you can arrange each of these eight things. But but by constraining it to the symmetries of the cube, it somehow creates some sort of hidden order amongst different variations. And so so that was the symmetries of the cube, which, again, you can only I don't think as an artist necessarily expects you to hear the the cube. But. Simon Stravinsky used to write, you know, actually, I can only be creative under huge constraints.
And I think a lot of composers love the constraints that mathematics gave them in which they confine. Then they can play and find a new sort of creative structures. Now, the interesting thing about Xenakis as well is that not only was he interested in mathematics and also a composer, but he's also an example of my my second art that I want to turn to, which is architecture. And he actually worked on a pavilion in Brussels.
And if you look at the plans for the pavilion, they look very much like that score for metastasis and fighting roots. He constructed this pavilion alongside my second choice of a secret mathematician from the world of architecture, which is Le Corbusier. You look obviously I worked with Xenakis on this this pavilion. But look, APJ was obsessed again with mathematics in particular.
He liked this idea of the Fibonacci numbers. The Fibonacci numbers were obviously embedded in the way that things grow naturally. So he believed that you should sort of mimic that in a building. And so he produced the series of numbers, his rouge and still his blue, which were used as his kind of numbers to the proportions that you should find within the building. So you can see after a while they settle down to this. They have exactly the same rule. You add 0.86 to 1.40 and you get to 2.26.
So this and he he thought that these are actually related. Something that goes back, of course, to Leonardo as well, that some of these Fibonacci numbers are related to proportions in the body. And this is a an idea that goes back to Vitruvius, the Roman architect thought that a building should somehow mimic also the proportions in a body.
And that's the sort of buildings that we all love. And actually, if you use these Fibonacci numbers, one of the reasons they seem to appear naturally in things that are growing in the natural world is the term you can use them very nicely to build up a structure. So if I take, say, a one by one room and add another one by room, one next to that, I'll have a one by two room.
Now I can add a two by two room because I know about the number two now on the side of that, but now I've got the dimension three so I can add a three by three room and you keep on adding these rooms. The Fibonacci of course one of the Fibonacci numbers and you get this natural spiral appearing and in fact this rectangle that's emerging is getting closer and closer to a rectangle that very many artists have found appealing, which is the rectangle which is in the proportions of the golden ratio.
So in the limits, this would be a rectangle which has this very special property that if you look at the, the ratio of a long side to the short side, if that's the same as the sum of the two sides to the long side, then this some rectangle is one that we seem to find most aesthetically pleasing. If I cut a square out of that, so then what I get is a rectangle which is also in the golden ratio. So it has a and a lot of canvases will often have these proportions.
Leonardo especially love these kind of portions. Also in architecture, if you take something like the Parthenon, the the ancient Greeks certainly knew about the golden ratio and it's believed that you can find golden ratios kind of hidden inside the proportions of the Parthenon. And in fact it's not only architects is. I did an extraordinary thing at the Royal Opera House. Looking at the mass behind the Magic Flute. And I discovered that the overture to the Magic Flute.
I don't know whether you know that. So there's something called the triple chord, which is Mozart's way of embedding kind of the idea of the Masons inside there. And when this triple chord occurs in the overture, is it exactly the moments where the golden ratio is? And Messiaen, Mozart must have known about this. It's too deliberate. So it's kind of 83 bars and you get this triple chord and then 130 bars after that. So there's a lot of examples of composers also putting key moments in a piece.
As this kind of proportion of the golden ratio. Debussy also did it, for example. But look at blues. You love this as well. And he felt that these movies in the series Blue gave rise to buildings, which had this sense of aesthetics inside it. One of the great examples is this one, which doesn't look a fantastically wonderful building from the outside, but if you look at the layout of the rooms on the inside, they're constructed according to these series rooms in this series Blue.
And I was talking to somebody recently, you know, somebody who lives here and says it's actually a beautiful building to live inside because of the way these rooms are laid out. Of course, look, appreciate in the first use the idea of proportions being important in the way that a building is built. Palladio, for example. PALLADIO Villas are so perfect in a way, because he was very mathematically sensitive. He liked kind of whole number ratios rather than these Fibonacci relationships.
And actually, it's very much related to music because if you take whole number ratios in music, you actually get notes with harmony on them. So here's a Palladio building, but I could put strings on the side and pluck them with give them to the links, the rooms. So in some way plant is buildings, as they are often called frozen music, because the proportions inside those villas are actually the proportions that we respond to musically as notes with harmony,
octaves and the perfect fit. It's intriguing that if you look at Paris, Le Corbusier, his sketchbooks and Palladio sketchbooks, they look very much like a mathematician sketchbooks. He is trying to find out all the different possibilities for the way a piece of geometry might work. Look, obviously, I love the idea of things with asymmetry. Palladio loved symmetry, but he's kind of the mathematical spirit at heart, at work, trying to see what different possibilities.
With these constraints, you can construct buildings. I mean, there are other wonderful examples of I mean, the Guggenheim in Bilbao, which is a Frank Gehry building, is almost like a piece of Riemannian geometry, which is a Corbusier chapel. It looks like a bit of hyperbolic geometry. And, you know, I think the modern skyline is full of mathematics these days.
And we have a project here in Oxford called Max in the City dot com where we've tried to record interesting examples of buildings with mathematics hidden inside them. And if you have any examples in your own city that you would like to add, it's a very interactive thing. We're trying to build up a kind of a sort of walking tours of the whole of the world, which viewing the world mathematically.
I mean, for one example, I think Zaha Hadid, who's been done so many buildings recently, the Olympics in particular, but Zaha Hadid studied mathematics in Iraq before she became an architect here in London. But it's interesting that I think this tension of the symmetry of the asymmetry Modular Man, I think, is kind of like a 20th century asymmetrical version of the Vitruvian Man that Leonardo drew. And in fact, the Vitruvian Man is a solution to an architecture problem.
Vitruvius left in those notes. He wrote about architecture, this kind of challenge that he believed that a building could encapsulate a human body, both in a square and in a circle. And a lot of artists try to solve this, and they would often put the square in the circle with a common centre, but they could never put a person inside it.
You've got these very sort of disproportion bodies when you stuck this in and it was Leonardo who's kind of solved this, realise you had to shift the centre of the square down off the centre of the circle. So you have different centres. The circle is centred on the bellybutton, the square somewhere else. But it interesting, I was kind of wanted in a way to choose.
Leonardo is my choice for my third secret mathematician from the world of the visual arts, because he so captures somebody who is a bridge between the sciences and the arts. He did so much sort of with these wonderful inventions that he made and fantastic art as well. But I went for a 20th century one as well. I sort of kept to that theme.
And so my choice was visual artist for my secret mathematician, actually Salvador Dali, because Salvador Dali's work just seems to be just so obsessed with ideas of science and mathematics threaded through them. In fact, he once wrote, I, I'm a carnivorous fish swimming in to waters, the cold water of art and the hot water of science. And it said that you always used to invite scientists round to his house, and he found them much more stimulating than artists for for generating new ideas.
And so you can see with lots of scientific ideas, there's DNA examples of DNA drawn inside the spiral of DNA relativity clearly inspired him with all his sort of watches falling off and things like that. There's always a lot of mathematics. So, for example, if you take the Sacrament of the Last Supper, he embeds that inside a dodecahedron. The symmetrical shape actually, that Plato believed was the shape of the universe.
And it's intriguing because this again harks back to artists who were using symmetrical shapes in the Renaissance. And in fact, the artists of the Renaissance were very helpful for mathematicians, because you'll see in this picture here, this is a portrait of a mathematician, Luca Pacioli. And you can see the dodecahedron again on the table here. But then there's another extraordinary symmetrical object in the top left hand corner, which is a it's a sort of glass structure.
And I think it's got a sort of field half filled with water as well. And this is a structure. It's made out of squares and triangles. It's actually called a round bead cube octahedron, and it's example of an all committee in solid. Now, you might have heard of platonic solids. There are these five platonic solids which make good dice the cube, the dodecahedron and three others.
But Archimedes just discovered that actually, if you don't stick to the face, it's all being the same symmetrical shape. So square a cube obviously has six squares, but. If you take something like a football, the classic football that you kick around on a sun that's made out of pentagons and hexagons. They all have the same length of edge, but you've got different shapes, but they're all arranged symmetrically on the side of that shape.
So the challenge was how many are the different shapes are there? And Archimedes discovered that there were 14 different shapes and. Q Including this wrong be cube octahedron. But his description of them was lost in antiquity. And it wasn't until the Renaissance that we kind of recovered what those 14 shapes were. And I think it is in part thanks to the artists who were working at the time with perspective, that this was the real challenge for them.
You know, can they draw these objects and sort of bring them alive? And Leonardo in particular, he illustrated a book by Pat Chorley, and it gives rise to lots of different examples of these. So. So I think here you see the artist and the scientist sort of helping each other in recovering what these 14 actually were. But Tony wasn't interested just in these very classical shapes. He was interested in shapes, these kind of modern shapes appearing in the 20th century.
So it's an amazing picture, the design of war, which pictures a skull. And then inside each of the sockets, the only sockets in the mouth, you have another skull. And inside there the hole was in there you've got another skull. And so you have this infinite regress. This is actually an example of a Lipinski gasket, a fractal with its infinite sums of complexity, and never get simpler as you go further and further inside the building, inside the picture.
And then there was another artist, Dali was doing this very deliberately. There was another very famous 20th century artist who was drawn to fractals, but totally intuitively and this is Jackson Pollock and now Jackson Pollock's paintings. I mean, he I think he's now been beaten now, but he held the record for the painting, which sold for the most amount of money in history.
But a lot of people said at the time, well, come on, my, my, my 210 year old twins could make this and just scatter the painting and you can make a Pollock. So what is it that Pollock was doing that was so special compared to what, you know, your kids might do with a pot of paint? Well, it turns out that what he was doing with something was something very special.
In fact, you can use a bit of mathematics to discover people started trying to fake Pollock because if they with so much money and you just need to scatter a bit of paint around but actually is quite difficult to fake Pollock because a Pollock has a very special property. This, this idea of a fractal that if you zoom in on a fractal you use the it doesn't get simpler, the complexity remains. So I've actually taken four different regions from one of Jackson Pollock's paintings.
One of them is the painting itself, and the other three are zoomed in portions of that painting. Now, I think you can probably tell that the top right hand corner is the most zoomed in one, but amongst the other three, I think it's pretty difficult to tell which is the original and which is a zoomed in region of it.
And so this is being used. The fact that if you look at the way the paints put on the canvas, it's possible to distinguish quite a lot of fakes that don't have this particular property. The reason Jackson Pollock was able to do this because is because the fractal is kind of the geometry of chaos. And the way that Pollock painted was very chaotic, I mean, in a mathematical sense. So he was apparently had incredibly bad balance and he used to often paint when he was drunk.
So the combination was such that, in fact, he created a chaotic pendulum, because if we if only was splattering paint, actually, I would have this is a fixed point and I would create a lot of regularity. But because Pollock was sort of not able to balance, particularly the pivot here was moving all the time. And he created this kind of chaotic pendulum, which when you look at the geometry that's associated with that, is this kind of fractal shape.
So there is a way to fake a Pollock, which is to set up a pot of paint on a string and then push it to make sure the top of the string is being pushed. Every now and again, you create a chaotic pendulum. So we actually did this in one of the BBC programs that I made. So this is so De Soto. Number one, we haven't quite perfected it. I didn't get anything on eBay for this, so I'm still working on it. So. But Pollock also. So Pollock was interesting.
We went to his studio and his studio is in Long Island. It's surrounded by all of these very fractal like trees. We went in the winter and you could see that he was sort of responding to the natural world around him because those trees have this fractal branching property and you can see you can even measure Pollock's kind of periods because of their sort of a fractal dimension.
And he he focuses in on a particular sort of fractal that is the one that is most resonant with the fractals in nature. Now, I'm going to come back to Dali because there are other examples of Dali's pictures where he was interested in geometry. Not that you can see around us in the physical world, but actually sort of beyond the physical world. So. He was very interested in four dimensional geometry, and here's an example of a crucifixion.
So he was a very spiritual man as well. So the idea of the fourth dimension was great for him, something that transcended the physical world. Somehow the fourth dimension had a spiritual side to it. So he did this crucifixion on a so this is actually a four dimensional cube unwrapped into three dimensions.
So if you think about how you would make a three dimensional cube out of a piece of paper, you would have six squares that you would cut out in cross shape, and then you would follow them up and you could make your cube. Well, it's the same principle at work here, because this is eight three dimensional cube stacked four on top of each other and four on the side. If you were living in four dimensions, you'd be able to wrap this net, this three dimensional net up to make a four dimensional cube.
And obviously, we're not in four dimensions, but you can still see the unwrapped four dimensional cube, which is this two inch locking sort of cross shapes. So for Dani, this is wonderful, this idea of the Christ being crucified on this four dimensional cross unwrapped if cube unwrapped into three dimensions. And the idea of the fourth dimension was also very fascinating for my fourth secret mathematician who comes from the world of literature.
Now, I think literature is a little harder to find sort of connections to in mathematics and and sort of works of literature, although I suppose poetry is a very obvious place to look. But Borges, I think, is a great example of somebody who is, again, stimulated by the ideas of mathematics in the sort of short stories that he wrote. And there's one in particular that I really love, which is called the Library of Babel. And so if you haven't read it, I really recommend it's only ten pages long.
But in this story, he's really sort of exploring the ideas of paradox of infinity, the shape of space. And actually he's asking questions that were really intriguing scientists in the 20th century at the same time. So this is how the piece opens. He describes this library. There's a librarian who's in this library. He's trying to find out what the shape of his library is.
He describes the universe, which others call the library is composed of an infinite, of an indefinite and perhaps infinite number of hexagonal galleries from any one of the galleries. One can see internally the upper and lower floors. So it's actually looks like a beehive, all of these rooms, and they're kind of layers of the beehive, one on top of the other. And the librarian, through the short story, sort of starts to explore and tries to understand,
well, does this library go on forever or is it infinite? Could he ever know that or is it finite? But how would that work? And by the end, he actually comes up with a solution. I venture to suggest this solution to the ancient problem. The library is unlimited and cyclical. If an eternal traveller were to cross it in any direction after centuries, he would see the same volumes repeated in the same disorder.
And it's very intriguing because his solution is actually one of the solutions that we think maybe the shape of our universe. If you think about our universe, well, it's the universe infinite. Does it have a shape? Is it finite? Well, it's kind of funny. If it were finite, how does that work? Because, well, the ancient Greeks used to think it was somehow enclosed in some sort of glass ball of stars on that.
But then what's on the other side of that? You know, are we living in The Truman Show with a camera crew sort of looking in on us? Often I do feel like that's but but but the interesting thing is that story is a came up with a solution that mathematicians came up with as well because here's a universe. This is a smaller universe than our universe. It's a two dimensional universe. Some of you may have played and you won't play asteroids in there. Yes.
All the old people sitting around saying, my my son would not be seen dead playing this game. But it's a very beautiful illustration of how a finite universe. So the universe is just on the computer screen. It's finite, but it's unlimited. It doesn't have any walls. It's not The Truman Show. When you go off the left hand side of the screen, you reappear on the right hand side.
And if you go off the top of the screen, you reappear at the bottom. So it feels like this thing is just going on and on and on. But of course, it's finite. So this is rather like the description that the librarian arrives at at the end of the Library of Babel. And the reason is that this does have a shape. It's in fact, the universe is in the shape of a Taurus or a bagel or a doughnut.
So when you go off the top of the screen, what you're doing is actually going around the Taurus inside and background again. If you go off to the left, you're going round the outside. And in fact, we can illustrate the top and the bottom of the screen, essentially the same. You join them up, you go round, and the left and the right hand side of the screens are also the same. So you can join those up. What you get is this bagel shaped universe.
Well, we live in a three dimensional universe. The librarian in the Library of Babel is in a three dimensional universe. So. So what's happening there? Well, it could be the same sort of thing, actually, because, you know, suppose this is our universe. We've had the big bang. And it's got to this size and. There's nothing outside this lecture theatre. In fact, the rooms of this lecture theatre are rather similar to this.
So if you go out to the right hand side of the lecture theatre, you repair the left. So when you get out there and just come out here, when you go out the top of the building, top of the ceiling, you come in through the floor and then we go in another direction. So that's like the game of asteroids. So when you go out the screen here, you repair the back of the lecture theatre. Sam said that we've embedded mathematics in this building. You might find that I get out of this lecture theatre.
So that's what we want for our undergraduate. Yes. But so actually, what does this universe look like? Because it's very strange, because the light switch is going out, the back of my head is going through the screen here and then reappearing at the back of the lecture theatre. So actually I can see the back of my head over there. And then another copy of me and another copy of me. So actually, this universe is finite, but it doesn't have any walls that you bounce off.
And this is what the universe looks like. And this is a potential shape for all universe. If the universe is finite, but with no walls, then it could be what I mean. Again, if I put this in four dimensions, I could wrap it up and make a Taurus in four dimensions or a bagel in four dimensions. And in some sense, that's the solution that Gabor has, comes to the library made out of these hexagons going on internally, up and down, left and right.
But it's you come back whenever you go off, you come back to where you started at. And so this is one solution, but it's actually the heart of one of the great mathematical theorems that was proved in the last decade. The Poincaré conjecture, which some of you might have read about in the newspaper of proved a few years ago by a Russian mathematician, Grigori Perlman, who was awarded our version of the Nobel Prize, the Fields Medal. And he also it was one of the millennium problems.
It was one of the million dollar problems. What he did was actually to to make a list of all the possible shapes that the universe could be wrapped up in. I've given you one, but there are other possibilities. What are those possibilities? So that question that Boneheads Hands was asking, okay, what is the shape of the Library of Babel is actually at the heart of one of the problems. That is one of the great problems have been solved in the last century, really.
In fact, the library Babel was an inspiration for a project that I did where I because of this connection between mathematics and the arts, I get asked a lot to go and work with composers or choreographers to try and give them interesting ideas, some new structures that might might help them in their creative process. And so I ended up actually working on it's called the 19th step, actually after another Borges story.
But we were inspired by this idea of the Library of Babel. And so it's a piece of choreography which I ended up actually performing in. So this is a I'll show you a little bit of this. This is me dancing. The router encompassed construction of a hexagon, followed by a proof of the irrationality of the square root of three. So. I think that's enough of that. I think I must be a first for mathematics and dance, actually.
But actually, it was during that project that I learned about my fifth and final secret mathematician who comes from the world of choreography. Because I think choreography is a great example of sort of geometry in motion. And very often when a choreographer is trying to, it's quite an abstract world as well.
And so, for example, Rudolf Laban, who's my choice of a secret mathematician from the world of choreography, really developed a very mathematical language in order to be able to articulate what was happening in a piece of choreography. And he also used to make his dancers try and give them a sense of the geometry around them.
So he would always ask the dancer to think of the three dimensional shape that was surrounding them, rather like sort of three dimensional version of Vitruvian Man, in a way. So I said, Man is inclined to follow the connecting lines of the 12 corner points of an icosahedron with its movements travelling,
as it were, along an invisible network of paths. And so you can really tell when somebody is being trained in a lab and start a dance because they have this very you can see almost the shape emerging as they move that their limbs. Now actually the project that I teach, that piece of choreography has grown and actually become a piece of theatre.
So if you want to see the current version of this thing inspired by the Library of Babel, I'm actually something I'm starting work on next week, but it will be on at the Science Museum. It's become a piece of theatre working with an actress from complicity called X and Y. So I play X and my Victoria Google plays Y, but that's wanted at the Science Museum from the 10th of October to the 16th of October, and then it's going to the Manchester Science Festival after that.
So if you want to see the current state of this collaboration between art and mathematics, then head along to the Science Museum. Now I talk a lot about the way that artists use mathematical structures in their work, sometimes deliberately, sometimes drawn to it intuitively. But I think it works the other way round as well, because I believe that mathematics is equally as creative a process as the act of creating something in these artistic disciplines.
One of the books that my teacher when I was at school recommended that I read that made me fall in love with mathematics and realised that it was a bridge in a way between these two worlds of art and science. Was a book by G.H. Hardy called A Mathematician's Apology. And in that book, he really describes what it's like to be a mathematician. He says a mathematician, like a painter or a poet is a maker of patterns. I'm only interested in mathematics as a creative art.
And actually, Henry James Graham GREENE wrote that this book was the best description of being a creative artist after Henry James's diaries. And and I think that's very often what motivates us as a mathematician. It's not the utility of our subject. Symes describes how we're having to at the moment, tell the government what the impact of our work is. And certainly we have extraordinary impact on the world around us.
But I don't think that's the motivation for for most mathematicians in in this department. I think that we're just intrigued by discovering new, interesting structures, things that move us. When I try and create a new piece of mathematics, I want to, to and I present it at a lecture here to my fellow mathematicians, I want to surprise them. I want to take them on a journey to tell a story and to to show them something they've never seen before.
And I think, you know, if you take here's a particular example, one of my favourite little theorems from number theory due to ferma. This is like a wonderful piece of music, the proof of this fact. So I'm so firmly on proof that if you take a prime and you divide it by four and it has remained a one. So for example, 41 is a prime. I divide it by four, I get remain to one. Fermat proved. You can always write that as two square numbers added together.
So 41 is four squared plus five squared. Now they're infinitely many of these primes which have remained two one on division by four. An absolutely extraordinarily you can always prove that they can be written in square numbers. Now, that for me is absolutely extraordinary. What a surprise, what an earth of a prime number has got to do with these two square numbers.
And so as you read the proof, it's like two sort of musical themes which seem to have nothing to do with each other or two characters. And then you see them gradually through the story or through the proof, changing, mutating, until you suddenly realise, no, the two sides of the same equation. And and it's that sense of surprise that these two things are connected, which makes this such a beautiful kind of mathematical result.
And I think it's that which motivates us when we're when we're trying to choose what mathematics. I think there is a lot of choice. I here's one of my mathematical theorems which we're talking about in my impact report to the government. I don't think this will be particularly useful for anything. I don't know. That wasn't my motivation. What motivated me about this was a discovery.
I mean, I don't expect you to understand everything is up here. But what I discovered was a new symmetrical object. Which had very surprising properties, some surprising properties connected to something completely different to to where the symmetrical object was somehow created. It's the inside this symmetry were questions about solving equations called elliptic curves are actually one of the other millennium problems. These million dollar problems is about solving equations like y squared.
It was executed minus x. Can you find two numbers which which satisfy that equation? And now I can. I could get a computer to churn out new descriptions of symmetrical objects and just run the thing. I can get computed to churn out new mathematics. But that's not mathematics. That's like the monkey at the typewriter writing, you know, just. Just random words. I'm combining them. The act of a mathematician is to to be involved in the creative process.
So make a choice about what is exciting to talk about the drama involved in actually showing that these two things which look totally different, are actually related. And so I think that's the motivation for the mathematician and often those things will have an impact then on the physical world around us because what we respond to are things which are those structures which are sort of hidden in the in the natural world. I'm going to end with a quote.
I want you to think, is this a quote by an artist or is it a quote by a scientist to create consensus precisely and not making useless combinations? Invention is discernment choice. The sterile combinations do not even present themselves the mind of the inventor. Now put your hand up if you think that that's an artist talking about their creative process. How many people think as an artist talking out? A few votes.
Yeah. How many things the scientists talking about what they do that female scientists. How many people are not really sure. It could be either. Yes, I see. I have to put you in that sort of category now because it's sort of like, okay, I'm not quite sure now what? Maybe the word inventor gave it away because that's not usually a word that's associated with the creative arts. It was, in fact, only punk who was asked this question about what the shape of the universe could be, in a way.
But actually, Stravinsky used to call himself an inventor. He used to think he was inventing his pieces of music. But I think it really is that connection, that mathematics is this beautiful bridge because it has that creativity, yet it is trying to describe the physical world. And I think in the end I sort of did end up being both on the artistic side and the scientific side. And I found that it was mathematics that was my way, too, to unite these two. Thank you.