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Blueprints with Marcus du Sautoy
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In this conversation, Marcus Du Sautoy explores the intricate relationship between mathematics and various forms of art, including music, literature, and visual arts. He discusses how mathematical concepts such as prime numbers, symmetry, and randomness influence creative processes and artistic expressions. Through examples from renowned artists like Shakespeare and Dali, Du Sautoy illustrates how mathematics serves as a blueprint for understanding and creating art, while also emphasizing the emotional and aesthetic dimensions of both fields.
Takeaways
- Mathematics and art are deeply interconnected.
- The circle is fundamental to both mathematics and nature.
- Prime numbers are essential building blocks in mathematics.
- Music often employs mathematical structures for creativity.
- Shakespeare used prime numbers to disrupt rhythm.
- Symmetry plays a crucial role in both art and mathematics.
- Dali's work reflects his fascination with scientific ideas.
- Theatre allows for abstract exploration of mathematical concepts.
- Ambiguity is embraced in art but avoided in mathematics.
- Randomness can lead to unexpected creative outcomes.
Chapters
- 00:00 Blueprints of Mathematics and Art
- 02:35 Defining Creativity and Its Interplay
- 04:24 Mathematicians as Collaborators with Artists
- 07:17 The Fractal Nature of Jackson Pollock's Art
- 12:54 The Significance of Circles in Mathematics
- 16:31 Exploring the Mystery of Prime Numbers
- 19:52 The Role of Primes in Music Composition
- 28:01 Mathematics and the Structure of Music
- 29:00 The Mathematical Foundations of Music
- 31:50 Art and Mathematics: Dali's Exploration
- 38:56 Theatrical Structures and Mathematical Concepts
- 43:46 The Distinct Narratives of Numbers and Art
- 48:07 Symmetry and Randomness: Blueprints of Creativity
- 58:49 Exploring Creativity Through Mathematics
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Transcript
Introduction and Interplay of Mathematics and Creativity
00:00:00
Speaker
Welcome back to Breaking Math. I'm your host, Autumn Feneff, and today we're joined once again by a man that needs no introduction, Professor Marcus de Satoy. He holds the title of Oxford Simone Professor of the Public Understanding of Science.
00:00:15
Speaker
He's a mathematician, author, and someone who has been helping us rethink science. the role of mathematics in our daily lives.
Blueprints and Creativity in Art Forms
00:00:24
Speaker
This time, we're diving into his latest book, Blueprints, how mathematics shapes creativity, from Jackson Pollock's fractals to Shakespeare's use of prime numbers, from symmetry in Bach to randomness in modern design. Marcus shows us how the language of mathematics doesn't just describe the universe, it collaborates with creativity itself. Marcus, welcome back to Breaking Maths.
00:00:49
Speaker
Out of curiosity, why did you choose the metaphor of blueprints and how did you decide to capture all of these mathematics and art in this interplay?
00:01:01
Speaker
Yeah. Well, I guess that the central idea of the whole book is that I think that mathematics is the study of structure. I think that's a very good description of what i how I view mathematics. And and structure is also key to many artistic practices. um I think a lot of people would think that artists are just kind of emoting, either musically or visually, or but actually structure is incredibly important for them too. And so often the structures that they're interested in ah things that I recognize as kind of mathematical in nature.
00:01:40
Speaker
And I thought that the word blueprint sort of captured that because... It has the idea of some ah kind of framework i within which you can, you know, build many versions of something or many different ah things based on ah kind of some underlying principles. and So I quite like the error idea of Blueprint. I mean, Blueprint is...
00:02:04
Speaker
kind of very much at the heart of, for example, architecture or um product design. So I sort of thought that blueprint was quite a um ah sort of non-technical word, non-mathematical word, which sort of helped to underpin both kind of structures in mathematics and structures in the arts as well.
Creativity, Consciousness, and Mathematics
00:02:27
Speaker
So in that case, since blueprints are the study of structure, how would you define creativity and its interplay? Well, that's a very ah challenging, I think, question because creativity is quite a slippery word. um And, you know, I think it's a bit like consciousness. Trying to define consciousness is quite ah challenging. But um for me, actually, the two are quite related because I think creativity,
00:02:57
Speaker
is um kind of our species tool to try and understand um kind of our inner world, what it means to be human, our humanity's place within the natural world. And it's sort of um creativity...
00:03:13
Speaker
is I think going back to the psychologist Carl Rogers, who talked about creativity being humanity's best tool to understand um this kind of idea of consciousness and our own inner worlds and the inner worlds of others. So so I think that creativity um is about trying to understand consciousness our our place in the universe and i suppose you know that's um i think partly why mathematics and creativity are so mixed up because ah creativity um if it's trying to understand our place in the universe and our relationship to it and
00:03:53
Speaker
Creativity is very often associated with, you know, maybe expressing things in a musical or a visual way. But I think that because mathematics is at the heart of how the natural world works, we're going to get this kind of interesting ah kind of interplay between um the world of mathematics and the world of artistic
Collaboration Between Mathematicians and Artists
00:04:13
Speaker
creativity. So if we're intersecting mathematics and creativity, does that make mathematicians collaborators with artists and not just interpreters?
00:04:24
Speaker
Yeah, I think so. and I suppose, you know, I wrote this book partly um out of all of the collaborations that I've had um going back since I first started training as a mathematician. And, you know, I think this is a very personal book because it's sort of um talking about a lot of the stories of times I've been either in the theatre space with people or or the concert hall.
00:04:49
Speaker
And I think that, I suppose, secretly, perhaps I wanted to be these creative artists. um and That may may have been partly what this book is about, my fantasy that, you know, I would have loved to have been a composer or or an architect. um But I think what I've loved over the years of being a mathematician is that, um you know, I've kept those interests and those collaborations ah very live. And so um i think that when I am working with, say, a composer, I
00:05:20
Speaker
for example, have set up a centre at the Royal Northern College of Music here in Manchester um in the UK with a composer, we call it PRISM, it stands for Practice and Research and Science and Music, is that I think that, you know, it is very...
00:05:36
Speaker
ah but The work that I do at the heart of that is very much a collaboration between mathematician and composer. I mean, for example, ah with Emily Howard, the composer, we actually composed a string quartet together, which explores the connections between yeah idea of mathematical proof and musical composition that I feel...
00:05:58
Speaker
Very often a musical composition feels a bit like piecing together a mathematical proof that takes you on a journey and you arrive at somewhere new. um so So I think that I certainly see it as a collaboration. and and I suppose what's interesting is that You know, quite often when I talk to artists and begin the work we do together, um they're very sceptical about whether we're going to find a connection.
00:06:24
Speaker
and Some are less so. I think there are some who are very um obviously courting me as a mathematician because they know that I'm likely to have some interesting structures that will be interesting to stimulate their creativity. you know But often I'll find that, you know, those collaborations start with just not really knowing ah that they have a lot of mathematics at the heart of their practice.
00:06:48
Speaker
And as we work, they'll say, well, this is the thing I'm sort of interested in. And I'll go, well, but I recognize that. That's really fascinating because that's, for me, a mathematical structure. so um So it's quite exciting. I think one of the themes that comes out in the book quite a lot is um the...
00:07:06
Speaker
the interesting tension where is a creative artist aware of the fact that they're creating mathematics? And sometimes they are very deliberately, perhaps using mathematical ideas in their work.
00:07:20
Speaker
But I think sometimes what's more interesting is when a a creative artist sort of explores ah something for his own aesthetic value and through that process almost rediscovers or perhaps discovers for the first time ah mathematical ideas. So, you know, for example, um i have a chapter about fractals and Jackson Pollock is a great example of ah um an artist who wasn't aware that he was doing mathematics. But if you a analyze his paintings, um you find that they have a very...
00:07:53
Speaker
i sort of special mathematical signature, which is that the geometric patterns he makes by um this kind of drip painting technique, are actually fractal in quality.
00:08:05
Speaker
The fact that they kind of have infinite complexity. So think this is why a Jackson Pollock is so exciting. You stand in front of a Pollock And it's very hard to judge the scale. You don't know whether you're close or far away.
00:08:17
Speaker
and that's the the work the fractal characteristic is is doing. It's giving you a sense of almost falling into the painting. And we've actually used ah this signature of ah the fractal quality of his painting,
00:08:31
Speaker
to identify ah genuine Pollux versus fake Pollux, because it's actually quite hard to do. You think, well, surely a Pollux would be easy to make. I just flick paint around. um But that generally would create something which isn't fractal, because um our arm is a bit like a pendulum, and pendulums are very regular.
00:08:51
Speaker
But Jackson Pollux's painting style was very different. He used his arm all the way up to his ah shoulder and was moving that around, and and that creates something... and which we call a you know, double pendulum, a chaotic pendulum.
00:09:05
Speaker
And the signature of chaos it is the fractal. So his painting style produced something quite um
Mathematical Analysis of Jackson Pollock's Art
00:09:12
Speaker
special. and ah But he wasn't aware of that. So, you know, he he went, he died before anybody understood that the unique quality of his paintings are that they have this ah fractal geometric structure.
00:09:26
Speaker
And I think, you know, going back to the idea that, you know, you know, creativity is about understanding our place in the universe. ah This is exactly, we can see this in Pollock's work, because Pollock was, in a sense, capturing the fact that nature is fractal at heart, the way that nature grows things.
00:09:46
Speaker
um I went to Pollock's studio. It sits in the middle of a forest, and Is the rumor true that you have ah your background of your wallpaper on your computer as Jackson Pollock's car?
00:10:02
Speaker
and and then he flicks the plane around so the floor itself in his studio um is you know a missing pollock because it's just covered in drip painting um And so I took a a really nice low shot of the floor and I now have that as the the desktop to my computer. So, and it, you know, it's, it's all the paint that didn't go on the canvas. It went on the floor, but, and but, you know, he his, his studio is surrounded by trees, which are fractal in quality. You know, you have big branch, then it branches and smaller branches and, know,
00:10:36
Speaker
It has this kind of interesting sort of scaleless property where the small things look like copies of the big things. So I think that he was responding to the fact that he saw around him fractals. He didn't know that those were mathematical shapes, but he wanted to make an abstract version of that.
00:10:54
Speaker
And we have this mathematical measure of how fractal something is, this fractal dimension. um So one thinks of a line as one dimensional,
00:11:06
Speaker
a solid figure is two-dimensional, but fractals are sort of somewhere in between because they're not really a line, but they they're trying to fill ah the space, the canvas, but they're also not just, you know a block of paint. So we can actually measure, and I talk about this in the book, a way of measuring, you know, how fractal something is.
00:11:27
Speaker
And what's rather beautiful is that the paintings of Pollock that we love the most... are those with a fractal dimension which is closest to the fractal dimension of nature.
00:11:39
Speaker
If it's a little less, we find them too simplistic, too much, and it's too complex. And so one of the reasons that we're responding to Pollock's work is that he's sort of abstractly capturing nature.
00:11:52
Speaker
But we really needed this kind of code and the of mathematics to to truly sort of understand um what was going on. Yet Pollock didn't deliberately put that in. i take a Pollock and then I zoom in and I ask people, can you tell which is the original and which is... ah i mean, obviously, when you get up to a certain point, the fractal quality has to go because um you start to see the spots of paint. And and that's also another really interesting tension.
00:12:23
Speaker
um that comes out in the book quite often as well is that um ah often of the ah kind of Mathematics might be perfect, but the realization of it often is necessarily an approximation. I noticed ah that you go through the book and you end up talking about circles a lot and also some structural configurations behind the circle when building a sphere in real life. Can you tell us a little bit about that?
00:12:54
Speaker
Yeah, I think the circle is so interesting. I mean, I sort of, the first two blueprints are almost like the blueprints of number, prime numbers, and the blueprint of geometry. Do you think, you know the circle is somehow the beginning of it all?
00:13:09
Speaker
um And it is a structure that humanities has been obsessed with Really, ever since we've started creating, you get i mean, here in the UK, we have amazing examples of stone circles, um ah which are, you know, thousands of years old. And I think, again, you know, the circle is a shape that we were encountering all the time. ah the the the sun is circle the moon going from circle to disappearing and back to circle um the ripples in water mean i suppose even our eyes um are examples you know so if you're looking into the deep into somebody else's soul ah you're looking through a circle so again you know the circle is uh you know so fundamental to the natural world but you you can say
00:13:59
Speaker
um Yes, but those stone circles certainly aren't exact circles. i mean, they're an approximation to a circle. And um and that kind of leads you on into um whether there is anywhere a true circle. And, you know, we we now...
00:14:16
Speaker
believe the universe ah according to quantum physics is quantized in nature so everything comes in little bits it's kind of pixelated rather than ah you know continuous and that means that any attempt at doing a circle um will probably always be an approximation and i suppose that goes back to you know plato his idea um of And I talk about Plato quite a bit in the book because often his ideas are that, you know, there's the platonic realm where these structures can exist in the mind.
00:14:52
Speaker
And then, you know, what are we as humans? We are looking at the shadows in the cave of those structures and and and they will always be an approximation of the kind of pure mathematical shapes.
00:15:04
Speaker
And actually Plato was always rather dismissive of um art. in He felt that that was an approximation of the approximation, um that ah art was never going to capture the pew pure kind of beauty of the abstract mathematical realm. So it's interesting that that kind of tension about, um ah yeah and it it does sort of,
00:15:31
Speaker
ah highlight one of my big theses, which is I think that the universe is a physicalized piece of mathematics, that those structures of mathematics, the blueprints of mathematics, the circle, prime numbers, fractals, exist before ever the universe came into existence.
00:15:51
Speaker
um And, you know, maybe that's the kind of what one's trying to look for something very spiritual, um you know, that that mathematics is it is the beginning of it all.
00:16:04
Speaker
And why we are seeing so much mathematics whenever we try and kind of ah explore the the physical world around us. Because it is kind of a piece of mathematics at its heart. You also talk a lot about prime numbers in the book.
Prime Numbers in Art and Music
00:16:20
Speaker
And you share everything from stories on gear theory. And also, what's the story behind calling your twins 41 and 43? Yeah.
00:16:31
Speaker
Yeah, well, i I suppose prime numbers, you know, is part of my own mathematical research. So they're very much um something that I'm very passionate about. They're still a mystery. We don't understand these numbers.
00:16:43
Speaker
um And so ah let me give you an example of something we don't understand, um which is we know that there are infinitely many these indivisible numbers.
00:16:53
Speaker
So the primes are sort of like the periodic table of mathematics. They are what build all other numbers. um We know there are infinitely many of them, but ah what's curious is that every now and again, we see two numbers coming up, ah which are like both prime.
00:17:11
Speaker
I mean, obviously you have to be odd. um ah The only... um Even prime is number two. That's the odd one out. um So the closest two prime numbers can be together is like two apart. So, for example, 17 and 19 or 41 and 43. And ah we believe, but we don't know, this is still an open problem, that there are infinitely many of what are we call twin primes. so But we don't we don't know the answer to that. that's really um Primes get rarer and rarer as we go through the universe of numbers.
00:17:45
Speaker
Yet every now and again, so they should be kind of spreading out, but we think that infinitely often you'll keep on seeing these little twins. um So the reason I wanted to call my daughters 41 and 43 because they are identical twins and I thought that would be quite cute.
00:18:02
Speaker
um to name them after 41 and 43. But my wife put her foot down and said, to you no, we can't do that. There's a bit Elon Musk-y. um It was something quite American. but It's before Elon Musk was even on the scene. But um anyway, they're my secret names for them.
00:18:18
Speaker
um But no the reason they come up in the kind of the world of creativity is is very interesting because, as you said, you mentioned gears there. um One of the things you don't want very often with gears it' is that they wear each other down because you keep on getting um you know teeth meeting each other too often.
00:18:39
Speaker
And the point about primes and why they're quite often interesting from a sort of creative point of view is the primes allow you to keep things out of sync. um So, ah that ah means that ah they that, I mean, let me give you the example, which is actually the first story in the book, um which is about the French composer Olivier Messiaen.
00:19:03
Speaker
Because he has a wonderful piece, it's probably the most iconic piece of the 20th century called The Quartet for the End of Time, ah which is very famous because he wrote it whilst he was a prisoner of war in Stellag 8A during the Second World War.
00:19:16
Speaker
And the first performance was in the prisoner of war camp by him as a prisoner and three other prisoners. um ah So it performed, you know, in January 41, 40, I think. um But it was, yeah you know, an incredibly famous performance.
00:19:33
Speaker
um But what's so interesting is the piano part. This is the part that Messiaen was playing because he wanted to create this kind of weird effect of somehow something repeating, but you never kind of feel it repeats. And so what he does is he has a 17 note rhythm sequence.
00:19:49
Speaker
which does repeat itself over and over again. It sort of starts just crotchet, crotchet, crotchet, then goes into a lovely syncopated rhythm, crotchet, minim, then just repeats. So that could be very boring, but he has a 29 note harmonic sequence. So the chords that are played, 29 chords, and then they repeat, 29 chords.
00:20:07
Speaker
Again, could be very boring, but when you put the two together, something very interesting happens because Here's how the primes keep things out of sync. 17 and 29.
00:20:18
Speaker
So when the rhythm is finished, it starts again. The harmony hasn't finished. It's working its way through its 29 chords. Then the 29 chords start again, but the rhythm is somewhere completely different.
00:20:29
Speaker
So every time you hear the the chord sequence, it's played with a completely different rhythm. And just think of Messier and 30. that wouldn't work because immediately you'd have the 15 rhythms, 15 rhythms, and the chords, when they start again, would just repeat. So the thing just repeats very quickly.
00:20:50
Speaker
But the 17 and 29, Messiaen has used that to keep the whole thing just never quite repeating, yet it's built out of something with with this repetition. um So that's a beautiful way that Messiaen used the the kind of indivisibility of the primes to create this kind of musical effect.
00:21:11
Speaker
And here's another example. I did some work with a composer that worked with Olivier Messiaen, George Benjamin, and i I asked him, I'm very interested, Messiaen has so much interesting mathematics throughout his music, and actually Messiaen is a character that pops up in the book in quite a few different places in the world of symmetry. and um And here's my favorite one of my favourite composers.
00:21:34
Speaker
But George Benjamin said, no, he had no mathematical training at all, but he was drawn to these structures for their aesthetic value. So he almost discovered the importance of the indivisibility of primes through his musical exploration, which I think is kind of rather beautiful. and ah But there's a lovely way that these primes... I mean, I love this chapter, to The Primes, and I think it sort of captures one of the qualities of the book, which when I wrote it, I really enjoyed, which is um starting with a mathematical structure...
00:22:13
Speaker
understanding how different artistic practices kind of realize this structure in such different ways. So you've got it happening in music with Messier, but um when you come to architecture, these primes are used in acoustics.
00:22:29
Speaker
So a concert hall very often has prime dimensions, but ah precisely again to make sure things don't come into sync so when sounds bounce off uh the what what what you don't want in a concert hall is reverberation or dead places in the in the auditorium and the use of primes means that the sound waves uh sort of never kill each other they don't reinforce and you get a really big boom um but then you come to something like literature how are primes used in literature
00:23:02
Speaker
And I think this was actually one of my biggest surprises. that Doesn't Shakespeare use that as well? Yeah, Shakespeare. you would I mean, you would never expect โ everyone thinks of Shakespeare as a wordsmith. What on earth has he got to do with numbers?
00:23:16
Speaker
Oh, there's a lot. Yes, there is a lot, exactly. um ah But I think that was a ah surprise, actually, to me. yeah And I mean, there's obvious, because he's a poet, poetry has a lot of mathematical structure in it. But I think the surprise for me was how he was using these numbers. So everyone knows that Shakespeare writes in iambic pentameter, which is 10 syllables, five groups of two, ah short long, short, long, short, long, short, long, short, long.
00:23:48
Speaker
is a very, you know, that's a very recognisable that iambic pentameter is actually used by a lot of different poets at the time. um But when he wants to you to really take notice, he changes that. And instead of using 10, divisible number, fives and twos, he uses an 11-beat structure. So what's the most famous line in Shakespeare?
00:24:14
Speaker
To be or not to be? That is the question. If you count those up, it It's one more. It's 11. It's a prime number. And it's very... it kind of disrupts your sense of rhythm. 11 doesn't fall naturally.
00:24:27
Speaker
um and you know, he's using that to wake you up. You know, you're kind of soporically being lulled into a sense of um yeah sleepiness by iambic pentameter. And then he says, OK, no, but this is the most important speech...
00:24:45
Speaker
in the play. ah And he uses that 11. ah In contrast, um he uses the number seven whenever magic is being used. um So actually one of my favorite plays is Midsummer Night's Dream. It was something I actually performed in when I was a student in Oxford.
00:25:03
Speaker
And of course, there's lots of magic in Midsummer Night's Dream. uses clustering of 575 going throughout. ah Well, actually, the 575 is kind of interesting. I mean, actually, it's mostly sevens for magic. Okay.
00:25:18
Speaker
So, for example, here's Puck as he um takes the magic potion, which... gets all the lovers to fall in love with her the different people. So, Chal, upon thine eyes I throw, all the power this charm doth owe.
00:25:33
Speaker
When thou wakest, let love forbid, sleep his seat on thy eyelids. it's it it It goes down from that ten to seven. And and even when the you know the other place where there's magic, the witches in Macbeth, when shall we three meet again? And ah and these odd numbers were really...
00:25:54
Speaker
kind of challenging ah the king at the time ah ah during that period. ah um obviously Obviously, Elizabeth I was the principal ah um member of royalty for Shakespeare, but um there was a ban on using any poetry which had odd numbers in it.
00:26:14
Speaker
um And this was partly because, you know, like we we like something with an even rhythm to it. And so it felt very ah like a, um like ah ah somehow a carriage with a a leaky um ah wheel or something that it didn't feel right. So, and so it's quite a challenge.
00:26:35
Speaker
And, you know, Shakespeare... I it it's at the time, if you think about the period of Shakespeare, it's a really exciting time scientifically
Mathematics in Shakespeare's Work
00:26:43
Speaker
because you've got... um ah Shakespeare probably heard about the ideas of Galileo and Copernicus about the shift of the earth not being at the centre or the sun being at the centre. And I think if you look at the later plays, you can see quite a bit of influence of um those new scientific ideas um that I think Cymbeline, for example...
00:27:04
Speaker
ah You can really read as a statement of this change in our perspective. And there is some evidence that he probably was friends with John Dee, who was um the kind of court mathematician in the Elizabethan um ah period.
00:27:20
Speaker
And John Dee was one of the first to publish um um an edition of Euclid's Elements. So mathematics was certainly in the air. And I think Shakespeare...
00:27:32
Speaker
ah probably was doing that deliberately. Here's an example, I think, unlike the Messia and the Pollock of somebody who knew what they were doing. this wasn't This wasn't intuition. he I think he really understood how he was disrupting things. To shift this a little bit, can you go into some of the details on how music usually has a tempo of 4-4, but there are other folks that use 5-5 or some also regularities?
00:28:01
Speaker
Yeah. Well, I suppose of all of the ah creative arts, people often talk about this connection between mathematics and music as being the strongest um connection. and And I think i there are many reasons for that. um You've already kind of alluded to the idea of rhythm being one of them. But um ah I sometimes I gave a definition right at the beginning of mathematics being the study of structure.
00:28:27
Speaker
Another one I often give is mathematics is the science of patterns. And often say that I think that music is the art of patterns and the word pattern is really important to both practices. And I think that's why when you're listening to music, i mean, music is very abstract in its nature.
00:28:45
Speaker
um The way the brain kind of navigates the musical world is spotting these patterns and noticing when something has changed but is related to what you've just seen and suppose this is one of the ways ah that mathematics and music have a very close bond is that and there's something you see very often in the way that bach writes his music is that ah if you're doing like theme and variations you've got a a theme and then you want to somehow ah change it up a little bit, but you want that variation to be connected to the theme. So often a musician will use ideas of symmetry
00:29:24
Speaker
in order to to do those variations. So when a theme might go up, you might very often hear it kind of kind of cascade down, almost like in a mirror symmetry, or it might invert the thing in a different way or transpose it. So you see lots of ideas of symmetry at work, especially in Bach's music, in sort of that that idea of theme and variations.
00:29:47
Speaker
But it actually right back to the... ah the notes we use to make music have at their heart mathematics. And this goes right back to Pythagoras, who understood why do we find some things harmonic and other things discordant?
00:30:03
Speaker
And he started experimenting and understood that the notes that we like ah that we find harmonic actually have a very simple, fundamental mathematical relationship. So um if you play a note on a guitar, have a guitar behind me, um play a note on a guitar and then half the length of the string, the note almost sounds the same.
00:30:25
Speaker
um and we call this the octave. And ah the... the ear is responding to the fact that there's a one to two relationship between the wavelengths. ah The length and the half produces something which almost feels the same note.
00:30:39
Speaker
And then if you mix that up and take a third of the length of the string, then we get something called the perfect fifth, which is the building block of of all harmonic music. um And that's what Pythagoras understood. If you use these kind of harmonic pairs to build up music,
00:30:56
Speaker
It leads to the fact that this octave um and very mathematically naturally deserves to be divided up into 12 notes. So anyone who plays the piano, for example, will see these 12, pattern of 12, great, well white white and black notes, and you see it repeated. And you you might ask, well, why did we divide it up into 12 and not, you know, why didn't we go decimal and do 10?
00:31:22
Speaker
Well, there's a very mathematical reason because the air is responding to this, um the fact that we find two notes ah harmonically beautiful and interesting when they have this mathematical relationship. You also mentioned in the book that some of the artists like Salvador Dali reach for mathematics to explore spiritual or transcendent type art.
Salvador Dali and Mathematical Ideas
00:31:50
Speaker
yeah go Go into that a little bit because I found that fascinating. It's kind of the absolute opposite of Jackson Pollock. Yes. And I think it's a very good example of an artist who sought out mathematical ideas for their inspiration.
00:32:08
Speaker
Pollock discovered them intuitively. Dali, in contrast, was hugely inspired by spending time with um scientists. And he used to say, you know, I'd rather swim in the hot water of science than the cold water of art. um And he used to have salons where he would invite scientists around. and And and find out about what the ah current exciting ideas were. and so And you can see that very obviously in many of his paintings, his obsession with Einstein's theory of relativity and the the nature of ah the flexibility of time. So we see these ah ah clock faces that are sort of warping.
00:32:48
Speaker
He was very obsessed with the idea of DNA. um So we see DNA spirals ah running through his work. But the the mathematical things that I found very interesting were the that he was very interested in the idea of mathematicians talking about shapes in higher dimensions, in four dimensions. um And actually, a lot of artists found this kind of discussion really stimulating. Yeah.
00:33:17
Speaker
And you can see why. It's sort of, it's going into the realm of the imagination, it seems like. What, more you know, what does a four-dimensional cube look like? um So Darnie has this extraordinary painting, um which is, i so obviously we can't,
00:33:34
Speaker
make a four-dimensional cube because we live in three and three spatial dimensions, but just in the same way as a three-dimensional cube can be unwrapped into a two-dimensional net from which we can then fold up and make a three-dimensional cube, ah mathematicians understood that a four-dimensional cube can be unwrapped to a net um in three dimensions, which if you put it in the four-dimensional world, you could fold up to make a four-dimensional cube.
00:34:01
Speaker
Now, this net of a four-dimensional cube is rather beautiful because it's, um if you think about the net of a three-dimensional cube, what is it? It's a cross shape.
00:34:12
Speaker
It's four squares stacked on top of each other with two squares either side, and you fold those up and make a cube. So the net of a four-dimensional cube is a stack of four four three-dimensional cubes on top of each other, and then you have four cubes arranged around the the top of that um shape.
00:34:35
Speaker
For Darley, when he saw that, it's often given name given the name of a tesseract, he saw this unwrapped four-dimensional cube, and it looked like a cross. It looks like two intersecting crosses, basically.
00:34:48
Speaker
And so there's this beautiful painting that he did of Christ on the hypercube, this unwrapped four-dimensional cube.
00:34:58
Speaker
ah ah it's ah But ah for me, I think, you see, one of the things which was very important to Dali was his spiritual and religious side. So for Dali, I think this idea of a of a hidden...
00:35:11
Speaker
other dimension that we can't see so resonated with him for a spiritual dimension that the mathematics was being used there to sort of give a hint of, yes, we have these shapes, but there is something beyond that's um that we can't see. And for Dali, that was a kind of four dimensions for him. He could make sense as a spiritual dimension.
00:35:38
Speaker
The other um person who i who comes up quite a lot in the book as well, um an author that is one of my favourite authors is Borges. and And Borges, again, he read a lot about four-dimensional ah geometry, lot it read a lot about infinity. That comes a lot in in his short stories. um But I think that Borges, like Dali, didn't really have the...
00:36:01
Speaker
technical mathematical language to understand what these things were. and So he used his own language, which was the language of storytelling. Dali used his own language, which is the language of of painting.
00:36:14
Speaker
And i think this is so exciting, we you know, that although we're trying to understand our universe, but we find our own particular languages that are best suited for us to navigate that. And for Borges, it was storytelling.
00:36:29
Speaker
And so although he didn't quite understand what a four-dimensional shape was, and in one of his stories, he tells us about a library. He was actually a librarian at one stage in his life.
00:36:39
Speaker
And he talks about a library project And are the librarian's stuck inside this library. And the librarian wants to work out, what is the shape of my library?
00:36:51
Speaker
And he talks about the library being everything. There's nothing outside the library. The universe, the library is the universe. um So actually what Borges is doing in that story is trying to understand actually the shape of our universe.
00:37:04
Speaker
But he does it through this um more tangible thing of the library. the the library shape is described as like ah made out of the rooms are hexagons so it's kind of beehive and there are many ah floors going up and down but then Borges his librarian sort of starts to speculate well does it just go on forever or is there a a limit? Is there some point where I hit a wall? and But then that raises the question, what's behind the wall?
00:37:32
Speaker
But the universe, the library is everything. So so he he says it can't be that. And eventually comes up with this solution that maybe i when I travel sort of in one direction, eventually I come back to where I started. And perhaps when I travel out um in a different direction, I just keep on bank coming back to where I started. And even when I go up...
00:37:51
Speaker
through the the the floors above, I suddenly find myself and suddenly find myself returning to the the floor I started in. If you understand this and analyse it mathematically, Borges in that story has actually constructed a four-dimensional torus, a donut shape, you know, like Homer Simpson donut, but a four-dimensional one ah where somehow all of these loops are like going around the four-dimensional shape.
00:38:21
Speaker
and And that's what's so beautiful. You know, he's used this story to actually create um a four-dimensional, ah I mean, the library is kind of the surface of a four-dimensional torus. Amazing.
00:38:33
Speaker
Amazing. It absolutely
Arcadia and Chaos Theory
00:38:36
Speaker
is. And one other thing that I found very interesting in literature was talking about Tom Stoppard's Acadia.
00:38:45
Speaker
Oh, yes. And is it more well-suited to exploring mathematics concepts through structure rather than content? Well, ah I think...
00:38:59
Speaker
ah theatre is one of my other big passions and I think that theatre is a lovely space to kind of experiment and try things out that perhaps wouldn't work if you were trying to be too literal. So I think that um i I enjoy working with theatre companies a lot and I talk about that quite often in the book um because it's it's almost little more like a mathematical space than, ah than say film, which I think is like, has to be perhaps truer to reality than perhaps theater is. I mean, you can create, and I talk in in the book about um the amazing worlds that um the Marvel universe creates using fractals, for example.
00:39:41
Speaker
um But there's still a sort of, it has to be believably real. Whilst in the theater, you suspend your disbelief and can explore, I think things which are are more abstract in nature. and So Arcadia is kind of interesting because you mentioned structure and content. I think Arcadia is beautiful because it has both.
00:40:01
Speaker
Arcadia certainly has content talking about um the ideas of fractals and chaos theory, ah Thomasina, ah the young woman at the heart of the play,
00:40:14
Speaker
essentially discovers the ideas of chaos and fractals. But because she's not in the computer age, this is, you know, I think it's 18th century, it's set in, ah ah she doesn't have the tools to genuinely understand what's going on. and and we only really discovered um the ideas of fractals when we had a computer to experiment and just make things like the Mandelbrot set. um So certainly what's lovely is that a discussion of, ah for example,
00:40:45
Speaker
the impact that the butterfly effect might have on history. I mean, the the idea that, you know, just a small change in ah the narrative could have huge implications. And so Stoppard certainly embeds that into um the the story of Arcadia, that if one thing had changed, ah the story would have been completely different.
00:41:10
Speaker
But I think structurally, what I find most exciting... So the content is kind of the most obvious place to explore mathematical ideas. But Tom Stolvard, I think, goes one further because I think that he actually makes the play mathematically structural at heart and fractal.
00:41:27
Speaker
That what you see is, too, the play is... ah sort of two timeframes, 20th century and I think it's 18th century. um And the yeah the the yeah the historical story is mapped out over three years, whilst the 20th century story is over three days.
00:41:47
Speaker
So you already have this kind of scaling effect and and you just see repetition of ideas through through the historical period reflected and condensed in the um in the modern period. And so I think Stoppard is really genuinely sort of trying to make a mathematical chao a fractal structure to the play that has at its heart also the content. um Another play that I really love, which I think does this superbly, is Michael Frayne's Copenhagen, ah which is a play exploring, again, the idea of history and the idea that not about chaos theory, but about quantum physics, that every time you observe
00:42:28
Speaker
and try and understand is a scene from history, somehow every time you observe it, it could do something different. And so in Copenhagen, he he repeats this meeting between Heisenberg and a Bohr that happens in Copenhagen. And we never, i mean, historically, we don't know quite what happened at that meeting.
00:42:48
Speaker
And we don't know, was it Heisenberg warning Bohr about the nuclear program? Was it Heisenberg trying to get some help with the Nazi program? And what's lovely is that, you know, his ideas of quantum physics, which is at the heart of what Heisenberg and Bohr did, but Michael Frayn,
00:43:08
Speaker
he the play is a kind of experiment in quantum physics because he just you you repeat the scene over and over again and every time you do the scene it's different it's a bit like every every time i look at the electron it goes through a different slit um so i i love when a a playwright goes that one step further and not only has science or mathematics as content But can they actually realize it as structure? Now, what kind of stories do numbers actually tell us versus some of the stories that artists will tell us? Is there, i know that there's a lot of crossover, but where are these distinct differences? Both are perfectly imperfect in their own sense.
00:43:50
Speaker
One of the words that, for me, became quite important in distinguishing, for example, the mathematician's practice versus the artist's practice, perhaps is the word ambiguity. You know, ambiguity for a mathematician is anathema. We like to really pin things down and and for you to for us all to understand simultaneously that there are infinitely many primes. it There shouldn't be any question at the end of telling that story that everyone understands, oh yeah, there are infinitely many primes. um
00:44:23
Speaker
Yet for an artist, I think ambiguity is quite an important, that they they want to allow... ah space for each person encountering their work to bring something of their but of their own to the the story. So there has to be sort of perhaps that room. And often, you know, artists will hide the way they've used a structure.
00:44:48
Speaker
So um they and they will want to challenge you maybe with um ah trying to kind of um um pick you know why is this thing working so brilliantly? And and so um and they often will not reveal. Although, you know frankly, mathematicians can suffer from this as well. So i think Gauss very famously enjoyed um sort of pulling rabbits out of hats. And people go, how on earth did you come up with that mathematical idea? And he he very famously once said, an architect does not leave up the scaffolding once he's finished the building.
00:45:26
Speaker
The idea being you don't you don't you don't necessarily need to show how how you came up with the thing. um ah But on the other hand, you know, in some sense you do. i mean, a proof is all about um ah removing any question that this thing might be true or might be false. um Whilst I suppose as in in maybe the arts, that's that's more interesting to have that openness.
00:45:52
Speaker
there's always that little bit of mystery in both. Yeah. Well, yes. ah you know, for me the reason I come to my desk every morning to do mathematics is not the things I know or the things we've discovered, but the the mysteries which still remain.
00:46:09
Speaker
um So they are certainly, you know, a massive drive for me in my own creativity. And And I should say, you know, one of the important messages of this book is that ah mathematics actually has perhaps more in common with the creative arts than many people realize in the sense that it is hugely creative in nature. The way that we come up with our ah discoveries, our proofs requires a real kind of ah aesthetic, artistic mindset.
00:46:38
Speaker
um And I think, you know, that's that's important for me. I think you know Some people worry when you talk about maths and music, am I trying to take the emotion out of music and make this cold clinical maths? um Often I'm trying to do sort of the reverse of that, which is I'm trying to show that mathematics often in its nature is very emotional and and actually the reason we like some maths and maths ah call other things just boring arithmetic is because there's a very interesting story being told by ah the the mathematical journey. So um I think that ah yeah the, I mean, that relates to the motivation for doing mathematics is often trying to unravel these mysteries. um
00:47:20
Speaker
And I suppose, you know, that's probably the motivation of many artists is that the mystery is often, ah you know, the our fellow humans, are the mystery of where we've come from, what what's going to happen next, oh i death, life, um those mysteries are what drive, ah you know, many creative practices. So I think, ah you know, we're both driven by mystery as the reason for wanting to be creative.
00:47:50
Speaker
Now, this is ah a little more personal of a question for this, but if you had to pick one blueprint that best captures you both as a mathematician and lover of the arts, what would it be?
00:48:08
Speaker
That's um that like trying to... Yeah, that's really difficult. But I suppose two probably come to mind, partly because the two things which I'm obsessed with are prime numbers, which we talked a little bit about.
00:48:20
Speaker
But the other one is symmetry. um So i my and mathematical research is trying to understand um the world of symmetry, what symmetries could exist. And I think symmetry...
00:48:31
Speaker
ah It is an interesting blueprint which comes up lot in music. um Symmetry is very often, you know, you'll find it in the way that Bach develops his music or Messiaen created some very interesting new scales which are symmetrical at heart.
00:48:47
Speaker
You find it used so often, well, across the architectural world, you know, from Palladio, loved using symmetry and it's the reason we feel so...
00:48:59
Speaker
I think content where when we're in a Palladian building is we're responding to that symmetry. And and and again, here we see this connection, mathematics, the arts, and nature, because one of the things we find in nature is symmetry is so important to the way nature grows things. And so I think we're often responding to symmetry in the world of the arts because it resonates with the symmetries that we see in the natural world around us. Okay.
Randomness as a Creative Tool
00:49:29
Speaker
So there was one story that I noticed in the book about randomness and you were tiling the floor of your kitchen. Can you tell us more about randomness in everyday life or even what you ended up doing with the different tiles?
00:49:50
Speaker
Yeah, because this is these the blueprint that I finished the book with, which is um the blueprint of randomness, which almost is an anti-blueprint because everything else up to that point is sort of ah is about structure and randomness seems to be positively the opposite. It's about something having no structure.
00:50:08
Speaker
But what we've discovered um is that so Randomness does have a ah patterns and structure. and And one of the interesting questions is when is something random, when when is it not?
00:50:19
Speaker
um We have signatures for when something seems to be genuinely random. And one of the interesting signatures um is that often randomness likes to clump things together.
00:50:30
Speaker
um So, ah um you know, buses are random. And one of the reasons, you know, you're waiting for buses, a big gap, and then three come together is you're experiencing that randomness. um And there was a very interesting artist that i encountered when I was doing the work with visual artists, which is Gerhard Richter.
00:50:51
Speaker
Gerhard Richter had actually a piece, which is exploring um randomness as a tool for creativity. And what he did was to take um ah five by five grid and then just randomly pick colors from um a sack. He had 25 colors in there and would just randomly pick one and then the first square would be blue and then he'd put it back and then he'd pick another one and it'd be red. and then um And he created 196 these and
00:51:23
Speaker
Actually, I was asked by the curator of the gallery, which displayed these, to give a talk about this. And and I use it actually, you know, 196 is actually interesting because it enables you, that's enough data to to genuinely test was Richter truly being random or did he go, no, I don't quite like that.
00:51:43
Speaker
I'm going to change that, you know, I'm going to cheat and make that red-blue because it looks nicer. And I did a statistical analysis on Richter's canvases and um and very interestingly, it passed all the tests of randomness.
00:51:57
Speaker
for example, quite often you see three yellows in a row. And often people will see that and go, oh, that looks interesting. That seems to be, is that saying something? Or if there's a little cross figure, its reveals that we are so obsessed with finding patterns and meaning that even when something is random, we will try and give it meaning. So, um,
00:52:19
Speaker
ah Now, ah coming back to the tiles that you mentioned. So actually, they're not on the floor. They're on my wall. But um I was working with a an artist and ah we developed these beautiful tiles which have different colours on them and different dimensions. And my kitchen was the first place where we actually used these tiles. And so I thought, well, you know, one one could do something interesting, like ah some sort of interesting pattern or something.
00:52:45
Speaker
maybe spell out something. um But I thought, no, let's try and do a kind of Richter and do do them randomly around the wall. And so I laid them out. And when my wife came down to see how they were laid out, she said, oh, no, no, that's terrible. Look, you've got three reds together. I said, yeah, that's um what you get with randomness. She said, no, I can't have a kitchen with three red tiles all together. And she started sort of rearranging them. And um In the end, actually, our kitchen is a great example of what looks like it's random. But if you're a mathematician, you can actually read that and no, somebody has had influence on this because it's missing quite a few characteristics of what should be random.
00:53:27
Speaker
Very, very interesting. Whenever you think of that, you think of... Even if you're looking at graph theory, take a step back. You look at these little clusters, even if they are random, and sometimes you'll see a dyad or triad or a supernode of something. So wouldn't you always see that in randomness as well? Yeah, I think that's why randomness is quite interesting as a vehicle for artist to take them somewhere new, because that randomness might lead...
00:54:00
Speaker
um to something which is ah kind of ah which has more meaning in it. And then you can start to maybe develop that. So I think that's, i mean, the other book I wrote, which has a lot of exploration of creativity, is um The Creativity Code about the way that AI can be creative. And very often AI is using perhaps some randomness um components it in its generative process.
00:54:29
Speaker
um But that's fine because that could often stimulate an artist to go, you know, I'm going to use this thing and then it's offering me something that then takes me on ah on a different journey. So, um you know, I think that that's often how randomness is used as, you know, I've got stuck.
00:54:45
Speaker
Let's just mix things up, see what happens. But then, you know, you won't just stay with the randomness. The randomness will take you to the next step of something you perhaps weren't seeing without it the use of it.
00:54:59
Speaker
Now, was there anything that you found extremely interesting or that you didn't expect when researching the
Unexpected Discoveries in Art and Math
00:55:07
Speaker
book? I noticed in the back you have a very long bibliography.
00:55:11
Speaker
hu Yes, I mean, i and I did a lot of research and it's based on a lot of um the years that I spent working with artists. Actually, I start the book with another long list, which is the dramatis personae. So I actually...
00:55:27
Speaker
you know i After I finished the book, I realized, wow, what a journey this book takes you on through you know so many different periods of history, different sorts of artists. And so I thought it'd be fun just to list ah the journey that um people will go on. um So, ah yeah yeah, I think...
00:55:48
Speaker
I suppose one of the most exciting things was um ah studying a piece of um music by a Greek composer called Yanis Sinakis, who uses the symmetries of a cube um for...
00:56:04
Speaker
structuring this piece is kind of, it's a bit like a theme and variations piece. um But when I started to analyse what Zanarkis had done, he actually used a structure in symmetry that I'd never seen before.
00:56:18
Speaker
In fact, nobody has ever explored this idea before. um So that's what's really beautiful is that This isn't a one-way dialogue. It is a one-way process. This is a dialogue where often a creative artist can start to explore something which might stimulate me as a mathematician. So one of my research projects that I'm working on at the moment really has its genesis in the work I did trying to understand this solo cello piece by Yanis Tanakis. And that was a surprise because I thought, um you know, it would be all things I knew and and telling those stories. But
00:56:54
Speaker
But that ah sort of discovery that Xenakis was making through music has set me off on my own new direction of research. That's absolutely wonderful.
00:57:06
Speaker
Now, as we're wrapping up here, is there anything that you want listeners or viewers to take away from this conversation or even know about math?
Embracing the Mysteries of Mathematics
00:57:18
Speaker
um I think one of the things that I think is really important is not to expect to understand everything. I think that mathematics almost suffers from the fact that, you know, we we can show all the working, we can show all our steps. And sometimes that limits us in being able to sort of jump forward and try and understand just a a feeling for ah kind of something. So, so yeah you know, I'm not shy in this book about showing people some mathematics.
00:57:49
Speaker
But what I hope is that even when people don't understand so all the details, that they'll just... at least get some sort of feeling for what's going on. After all, I think when I first started reading Shakespeare, I didn't understand very much of it at all. But as I spent more time with it, it made more and more sense. But I still, that just from the very early readings of Shakespeare, you fall in love with the power of the English language um to to tell amazing stories. And so i think always what I hope is it's ah that people will
00:58:25
Speaker
will approach mathematics um in an open manner as if they're listening to a piece of music for the first time, which they will not understand all the details and they won't retain everything, but they'll get a feeling for what's going on. And hopefully it will inspire them to come back and and and spend more time trying to understand things in more detail.
00:58:44
Speaker
Wonderful. Thank you so much for coming on the show. And it was such a pleasure for having you. And that's it for today's episode of Breaking Math. A huge thank you to you, Professor DeSatoy, for joining us and sharing how mathematics weaves itself into music, art, theater, and even the tiles on your kitchen wall.
00:59:02
Speaker
And if today's episode sparked your curiosity, be sure to check out his latest book, Blueprints, and see how numbers and structures can transform the way you experience creativity.
00:59:13
Speaker
You can find more episodes, transcripts, and resources on breakingmath.io and connect with us across all social media platforms, which are in the caption below.
00:59:25
Speaker
Thanks for listening. And remember, math isn't just about numbers. It's about the patterns and mysteries that shape our world. Until next time, stay curious and stay informed.