Instant Genius - How mathematics shapes human creativity

Episode Date: May 4, 2025

It’s commonly believed that the arts and the sciences have little in common with each other. The distinction that’s most frequently made is that the arts are creative in nature whereas the sciences are logical. But this couldn’t be further from the truth. In this episode, we catch up with mathematician and author Marcus du Sautoy to talk about his latest book Blueprints: How mathematics shapes creativity. He tells us how, fundamentally, mathematics is the study of patterns, structure and symmetry, how these patterns are found everywhere in music, visual art and architecture, and why we should be teaching students how to spot them in their everyday lives. Learn more about your ad choices. Visit podcastchoices.com/adchoices

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This is a job for Indeed sponsored jobs. This podcast is sponsored by name, audio, and focal. Streaming has made music more accessible than ever, but true listening is about more than ease. It's about quality. British audio experts name audio, alongside French acoustic specialist, combine handcrafted tradition with cutting-edge innovation and high-end materials, delivering digital precision with analog warmth.

So you can experience exceptional sound at home. Music just as the artist intended. Visit name audio.com to learn more. Hello and welcome to Instant Genius, a bite-size master class in podcast form. Every Monday and Friday you'll hear world-leading scientists and experts talking about the most fascinating ideas in science and technology today. today. I'm Jason Goodyear, commissioning editor at BBC Science Focus. It's commonly believed that the arts and the sciences

have little in common with each other. The distinction that's most frequently made is that the arts are creative in nature, whereas the sciences are logical. But this couldn't be further from the truth. In this episode, we catch up with mathematician and author Marcus DeSautoy to talk

about his latest book, Blueprints, how mathematics shapes creativity. He tells us how fundamentally mathematics is a study of patterns, structure and symmetry, how these patterns are found everywhere in music, visual art and architecture, and why we should be teaching students how to spot them in their everyday lives. So welcome to the podcast. Thanks very much for joining us. Thanks for having me on.

So today we're talking about your book, Blueprints, how mathematics shapes creativity. So let's start with the sort of big overarching question. I mean, there's a sort of common idea that science and maths are separate from art, maybe perpetuated by lots of things like the left brain, right brain myth. And personally, this is something that's always really confused me, because I think pure mathematicians and theoretical physicists are some of the most free-thinking, creative people that we have in the world. So where do you think this idea comes from?

Yes, I think it's the idea that we're after some sort of fundamental truths and somehow you can't be creative that you're being boxed in by this attempt to kind of just explain the universe around us. But actually to come up to with these solutions that mathematics and physics and science more generally has arrived at requires a lot of imagination, creativity, leaps into the unknown. And I think on the counter side, you see, I think that many people feel like artists are so free, they can do anything. And actually, that's not true. And most artists talk about the necessity of having constraints in order to kind of frame their creativity. So, for example,

Stravinsky always used to say, I can only be creative under huge constraints. And very often those constraints are, you know, structures that they put on themselves within which they then duck and weave and find kind of their creativity, but through these constraints. I think the blank page, you see, is just terrifying. But if you put some sort of structure on that blank page, then it kind of guides you into new realms that you may not have thought about. Now, I think this is really the thesis of my book in a way is that the connection, therefore, between the creative arts, and in particular mathematics, is this idea of structure. Structures are absolutely fundamental to what any architect or a composer is doing.

But I would define mathematics as the study of structure. You see, I think most people think that mathematics is all about arithmetic and long division to lots of decimal places. But that's like saying that music is just about scales and arpeggios. Yeah, they are somehow some of the ingredients for what we make. But actually, you know, I don't do very much sort of arithmetic in my day-to-day life as a mathematician. What I'm interested in is understanding and exploring the possibility of new structures. Yeah, so these days we have calculators and computers for the donkey work, don't we?

Yeah, exactly. And that never really was what mathematics was about for me. I mean, I think, you know, I would say that we have this kind of misconception about mathematics because mathematics is all about the kind of technical, utilitarian ideas about helping you to fill in your tax form and things like that. And, you know, I kind of wish that we had in our education system almost two mathematics courses. One is the kind of mathematical literacy, you know, like English literature. But then we do English language in our schools and we learn about Shakespeare and romantic poetry and George Orwell. And I somehow wish we had a kind of. mathematics literature course where we could explore some of these kind of big ideas of infinity

or geometries beyond our three-dimensional world, things which really have stimulated a lot of creative artists when they've encountered these. I mean, if you take someone like Borges, a wonderful Argentinian writer, you know, he got exposed to ideas of infinity, of geometries beyond our three dimensions, sort of, and these were somehow the catalyst for some of his wonderful short stories. He didn't really understand the mathematics, but he wanted to use his own language, which is the language of storytelling, to explore these mind-blowing ideas. So I sort of somehow think we're cheating our students at school of these wonderful, beautiful mathematical structures, which often are the catalyst for many creative artists. I completely agree. I thought an

overarching theme of the book was that mathematics really, in essence, is the language of nature. Exactly. And that's, in a sense, where the connection is, because I would say creative artists, you see, are using their language to interpret the natural world around them. Very often, if you take somebody like Jackson Pollock, his drip paintings are actually a response

to the kind of the natural world around him, and he represented it in this abstract form. But what he understands intuitively is that he arrives at sort of geometric structures in his artwork that I recognize as a mathematician, which is the idea of fractals. And fractals are, of course, how nature grows things. This kind of idea of a simple algorithm, which, if you keep on repeating, it gives rise to huge complexity. So there's, I would say the blueprint or the shape that's behind my book is the idea of a triangle with nature at one corner and then mathematics and the creative arts of the other two corners. And that we're both responding to

their natural world around us. But we find our own different languages to do that. Maybe it's the language of music, the language of visuals or poetry or mathematics. But that's why I feel like we're inevitably going to be interested in similar sorts of structures because they very often have their common root in nature. So let's, you mentioned music there. So let's kick off with that because that's one of my gems. It's just absolutely bound up in mathematics. and physics. So I think a lot of people don't know, like you have the fifth and full, boom, boom, and you talk a lot in the book about circles. So a lot of people that have studied music will know the circle of fifths, you know, with C, G, D, A, whatever. But this is really, really

interesting because our tuning system isn't correct mathematically. It's not the three to two ratio, you know, like, what can we say about that? Like, it goes all the way back to Pythagoras. Yes, I mean, the beginning is the kind of discovery that music is at its heart incredibly mathematical, and that's what Pythagoras discovered that, you know, why do we find this thing that we call a perfect fifth, which you sang so beautifully, you know, so harmonic. And this is universal across the globe. I mean, it's not just something specific to Western music. And Indian music is still based on this perfect fifth as being. somehow the seed of harmony.

And this is the beautiful thing. You know, why do we divide our scale? If anyone's got a piano at home, you'll see that there are 12 notes which repeat themselves. Why is 12 the division that we make of the octave? Why didn't we go decimal? Why didn't we do 10 notes when we introduce metric? It's because, as you say, this interesting thing about a circle, which we call the cycle of fifths. If you, I mean, we produce this two harmonic notes, which are in this three to two relationship.

And this is what Pythagoras discovered that that beautiful harmonic pair, if you take a guitar string and you'd play sort of half the length and the third the length, you get this pair, which is very harmonic. So what our ear is responding to, we're aesthetically being drawn to something with these beautiful mathematical relationship. But then if you then take this, you know, the top note of that first pair and make that the bottom note of a new. pair and you keep on doing this and you're producing more and more notes. Now, it could go on forever, but rather magically after 12 notes, we seem to get back to almost the note we started with. It's seven octaves higher. And so it seems to come in some sort of full circle. And those 12 notes that we produced, we kept on going. We could just get another 12 similar notes of just octave difference. So that's the reason that across the globe, we find the octave, which is in a one to two,

two relationship being divided up into these 12 notes. But as you say, it is actually exact. And there's this wonderful tension between, yeah, it's not quite joining up. And that difference is actually was the beginning of a lot of headaches about how to tune a piano such you can change, you know, from one scale to another. And I suppose that's also something quite interesting attention which arises in the book, which is this tension between the beautiful abstract world of mathematics and the physical realization of it. So even the circle, which is a very simple thing to define mathematically, it's beginning of Euclid's elements, is almost the building block of geometry, a curve which is an equal distance from a fixed point. But is there physically anywhere in the universe a true

circle? And quantum physics says actually, well, hold on, but our whole world is pixelated. If you go down to a particular, you know, magnification, everything becomes little pixels. And therefore we can't have this kind of perfect idea of a circle. At some point, we suddenly see the pixelisation and it isn't exact. So in a weird way, there's always this tension between the abstract, sort of putable world of mathematics and the rather messy physicalisation of it in the natural world around us. So let's stick with music for a moment, because you talk in the book a lot about Bark, specifically the Goldberg

variations, which is one of my favourite pieces. I watched Vick and Goa or Lufsen play it last month, in fact. And we've talked about the circular notion of notes or scales. But how about the circular notion of composition? Yes, I think this is really important because I think there's an obvious connection between mathematics and music on this kind of harmonic level

and rhythmic level as well. You see a lot of numbers being used to create interesting rhythmic structures. But I think what we're missing very often is that there's mathematical structure in the way that composers put these ingredients together to make a piece. And Bach is the perfect example. And I've done a lot of events with Pienist about the Goldberg Variations because there's just so much beautiful structure in there. So, you know, the Goldberg Variation starts with this beautiful aria. Then it goes on this incredible journey of 30 variations, and then comes back to the aria at the end. And so you've already got this idea of something coming full circle and almost like, are we starting again?

And even if you look at the 16th variation, which is halfway through this circle, Bach calls this an overture, which is generally the beginning of a piece. So Bach's almost teasing us, you know, where is the beginning of a circle? Actually, a circle doesn't have a beginning. But then you get another kind of structural circle appearing. in the canons. Every third variation is a canon where people probably remember this from school, where you start half the class starts singing, and then a little later, the second half joins in with the same tune, but delayed in time. So Bach loves the idea of a canon, but he

really plays with this idea that, well, you could get the second voice coming in, and perhaps a note higher. And so each canon through the variations, it isn't just a simple repetition. It's a repetition, but a movement in pitch as well. So gradually as you go through, each canon you hear the second voice starting higher and higher and higher until the eighth canon, you get this kind of joining up

because we hit the octave. And the octave is something which sounds almost like the beginning again. It's higher, but yet sounds the same. And so we give it the same name in our musical notation. And so we get this, within this circle, we get another circle,

which is a kind of harmonic or pitch circle. So I like to think that the Goldberg variations almost has the structure of a circle's worth of circles, which in mathematics we call a torus. If you move a circle around a circle, you get the idea of a torus. So there are lots of games and also the variations themselves. Bach is using ideas of another implausent blueprint that I talk about in the book, which is the idea of symmetry. So symmetry allows you to take a shape and then move it in some way such as it's related to what you started with. but somehow different. And so we see Bach doing this in the variations.

You'll have the first theme racing upwards, and then he'll reflect that in the horizontal line, and you'll hear the same theme but descended going downwards. It's the same theme, but reflected in some way. So Bach was a master using the kind of ideas of symmetry as a kind of algorithm for generating kind of complexity within music using simple ingredients. For maybe people who don't know much about music, essentially what we're saying there is mathematics, geometry, symmetry is absolutely everywhere. It's absolutely ingrained in us.

Absolutely. And I think, you know, what the brain is doing when we're listening to music and how we distinguish music from, say, noise, is the idea of some structure that we can, a pattern that we are recognising within that structure. And I suppose that's another definition I might give of mathematics, that it's the science of patterns. Patents, you know, we're a pattern search. So I think that's one thing where, you know, the brain is constantly trying to find patterns in the kind of chaotic world around us. And, you know, even it's how we survived in an evolutionary way.

Those who understood pattern and structure and symmetry were the ones who survived in this world. So the brain is somehow tuned to try and pull out pattern, whatever it's listening to a, watching or reading. And those patterns are often very mathematical in nature. Bonjour, compadre. It's the... Priceline negotiator! How do I negotiate so many great travel deals?

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and unforgettable listening experiences at home. Try it for yourself at a focal powered by name boutique. Visit focal powered by name.com for more information. So let's move on to visual art then. So I think a lot of people will have heard, it's on the cover of your book, the golden ratio. So what can we say about that? Yeah, the golden ratio is probably the fundamental kind of mathematical structure

which we see the artists being drawn to time and again because it's a kind of ratio to lengths, for example. You can define it in this way. If you have a rectangle where the long side of the rectangle, the ratio of that to the short side, it's the same as the ratio of the sum of the two sides to the long side. This is a rectangle we call in the golden ratio. And this seems to be something that we're aesthetically drawn to time and again,

both visually and actually orally as well. There are some wonderful examples of composers like Debussy or Bar, or even Mozart, using this kind of ratio in their pieces such that the sort of climax of a piece will come at the golden ratio distance from the beginning to the end. But visually we see this, for example, especially in someone like Leonardo da Vinci. He loves using this proportion. And I think very knowingly, in the case of Leonardo, he was very much trained as a mathematician as well as an artist.

I think this is the moment, you know, where we see these two. disciplines really fusing together in such an exciting way. And, you know, since then, we've seen kind of art and science somehow separating. But in this period, in the Renaissance, it's, they're really overlapping. And you see Leonardo working alongside mathematicians and artists creating the idea of perspective and things. But there's something rather magical about the golden ratio, which is it kind of focuses our attention, not on the center of the rectangle, but sort of four different regions where you get a kind of spiral, there's a very natural way to build a spiral into a golden ratio, because if you take a square out of that rectangle, whose dimensions are the sort of

shorter side, the rectangle that remains, a smaller rectangle, is again in the golden ratio. And if you take a square out of that, you then get a smaller rectangle, which is again golden ratio. So there's something very nice about this, because if you keep on dividing this shape, it somehow replicates itself at smaller and smaller scale. And this is something, for example, that Lecobuzier was drawn to, an architect who loved using the idea of the golden ratio in the proportions of his buildings, because he loved the idea that somehow you could take it a square out of the dimensions of a building or a room, and the dimensions that were left would be the same ratio, but scaled down from the original thing.

So he found that very attractive. And I think the key thing here is that the golden ratio is something we see time and again appearing not just in art, but in nature. So nature uses the golden ratio and somehow some fundamental numbers called the Fibonacci numbers, which are very much related to the golden ratio, in the way that it builds things. And so when we see shapes in the golden ratio or build out of these things called the Fibonacci numbers, we're actually responding to them kind of positively, I think, because, they are resonating with things in the natural world. And those are always things we, that some sense gives us a sense of pleasure and happiness and because, yeah, responding to the natural world is something that we've been doing ever since we've evolved as a species. So you mentioned the Fibonacci numbers. So let's have a look at that. So first off,

once you know them, they're everywhere, like petals and flowers, etc., etc. But what exactly are they? Yeah, so there are a sequence of numbers which starts one, one, two, three, five, eight, thirteen, and maybe you've spotted a pattern about how to get the next one, because what you do is you add eight and thirteen to get twenty-one. And so these have a natural sense of growth in them. You take the two previous numbers in the sequence to get the next number along. And this is why we see them somehow in the way nature grows things, because nature needs to use what it's made before in order to make its next step. And so, for example, as you mentioned, petals on flowers invariably are Fibonacci numbers.

But the lovely thing is, so Fibonacci, who is, who are they named after, an Italian mathematician of the 12th, 13th century, who discovered this idea that they are all over the natural world. But what I love is that they were originally discovered, not by mathematicians or scientists, but by musicians. and poets. So the poets and musicians in India, several centuries earlier, are interested in creating interesting rhythm patterns, especially, say, a tabla player who is, you know, knocking out rhythms, long and short beats, and wants to show off and produce as many interesting rhythms as possible out of these combinations of long and short beats. And as they discovered, you know, the way to build these rhythms, all the different possibilities, it turns out it's the Fibonacci numbers and this idea of using the kind of two previous sets of rhythms to make the

next rhythms along is why Indian musicians and poets should really be credited with the discovery of these numbers as the way to generate interesting kind of rhythmic structures. So probably they should be called the Hamashandra numbers rather than the Fibonacci numbers. But I think there's something really important here because, you know, I think often we think, oh yeah, of course the mathematicians discover these structures first and then maybe the musicians kind of plunder the cabinet of wonders that we have. But that's not true. Very often we see musicians being drawn to these structures for their aesthetic value. And then we mathematicians recognize them perhaps at a later stage as important kind of ingredients for making the natural world around us. So again, there's a real

sort of dialogue going on between the mathematicians and the creative artists and nature in discovering these structures. I think that's a really important point because a lot of people will think like whether they study maths at school or what's the point in this, what am I going to do with it? But it all comes from nature

and it's a way that we can understand the things around us. And that's very important. And like you say, I mean, it's not taught in that way. Like you say English language, English literature. And that's a crying shame to me. I agree.

I think that as soon as, you show somebody that nature is doing mathematics, you realize this thing is of fundamental importance. And I think, you know, one of the tragedies is that we focus too much on utility in our education system and less on just enjoying the beauty of what we have around us. And, you know, I think you see, understanding these structures allows you to build such interesting new things. I mean, I think that's, you know, where mathematics appears is first understanding. understanding the natural world, but then perhaps making sort of the human world around us. So the idea of, you know, we go back 5,000 years to the ancient Egyptians, the pyramids,

to be able to build a pyramid, you have to understand something about the geometry of that shape. But once you do, you have the power to create something really stunning, which, you know, is a mark of human creativity. So how do you think, sort of overridingly, how does, mathematics and aesthetics, how do they correspond, I suppose I'd say? Yes, I mean, we do talk a lot about beauty in mathematics, and it being an important quality of discovering truth, actually. I mean, Keats always talked about this connection between truth and beauty, and maybe, you know, aesthetics, where does aesthetics come from?

it probably comes from a kind of evolutionary value that having a kind of aesthetic response to something, a feeling of beauty is probably connected to an understanding of the natural world around us. So probably the structures that we find interesting mathematically and those that we're drawn to from an artistic sense and we call maybe beautiful or interesting or surprising are those which have helped us to navigate and understand our place in the natural world. around us. But you know, beauty and aesthetics isn't everything. I mean, sometimes there's something quite nice about the gritty nature of something which isn't necessarily so beautiful. So one of the

interesting kind of blueprints I talked about structures towards the end is the idea of randomness, which is actually, you might think is kind of like an anti-structure, an anti-blueprints, almost going against that sense of wanting to make something with structure. But again, you know, randomness is an important part of the natural world around us. We see maybe a kind of simple algorithm helping us to grow the things in the garden, but the influence of the weather will introduce an element of randomness, which means that everything doesn't look, you know, like a computer generated. And so introducing that idea of randomness, it's a key part of the natural world. And it's a key part of many artistic disciplines is introducing the role of kind of randomness, you know, whether it's in

writing or visuals or music does send you off in an interesting direction. But randomness itself has kind of structural elements and that's why we do have something called the mathematics of randomness which allows us to navigate our way through a world full of risk and chance and uncertainty. Yeah, this is really interesting. So people throw around the word random all the time. But, you know, what does it actually mean? Yes. It's very, Good question. I mean, we could do a whole podcast connected to, you know, is there anything truly random? Because in general, randomness is more a measure of our lack of knowledge of a situation. If you think about how a dice rolls, if we had all the parameters precisely,

then we'd probably be able to use Newton's equations of motion to predict how the dice will am. But that's the point we discovered in the beginning of the 20th century, this idea of chaos theory, where even if something does have a set of equations which are controlling it, a small perturbation can send it off in a completely different direction. And so, you know, I think that's one of the other interesting ingredients that artists use is setting up still something with rules, but something which is essentially chaotic in nature, and that a small perturbation can send the thing off in a completely different direction. And that's quite exciting. So it's that tension between something which has a rule, yet is still unpredictable. You probably have a problem. You probably

have to go to really down to the fundamental particles of nature to see something which we believe is genuinely random, which is quantum physics, says that essentially the world is random in nature, that there is no determinism in where you will find an electron. It is just probabilistic in nature. So we've talked about an awful lot there. I don't know. Personally, I'd like to get people enthusiastic about maths, anyone listening, you know, how can we do that? I think that understanding that things you are passionate about and love, for example, you're passionate about music or the visual world or maybe poetry or literature. What I'm hoping this book will do is to illustrate that behind many of the things you love are these interesting mathematical structures, even something like

Shakespeare. You know, Shakespeare you think is a wordsmith, but also love to love. to use number to create certain effects. So, you know, we know things come in tens in Shakespeare, riamic pentameter, but actually the most famous line in Shakespeare, to be or not to be, that is the question, is 11 syllables, a prime number. It's something which disrupts you. It sends you out of that sense of complacency and rhythm

because Shakespeare wants to indicate something that you should be alert to. And whenever there's magic in Shakespeare, he goes down to seven syllables. So you can understand that something magical is afoot through the numbers. So for me, those people who perhaps wouldn't normally think of themselves as interested in mathematics, if they understand that there are mathematical structures at work in the things that they love, in particular that they might be structures that will help them in their own creativity, then I think that's the key to them being interested to explore this. beautiful world of mathematics.

Thank you for listening to this episode of Instant Genius brought to you from the team behind BBC Science FACUS. That was Marcus Sautoy. To discover more about the topics we've just discussed, check out his book, Blueprints, How Mathematics Shapes Creativity. If you liked what you just heard, then please do consider subscribing to Instant Genius

on your preferred podcast platform. If you'd like to see our guests and presenters in person, then please also check out our YouTube channel. at ScienceFocus. The current issue of BBC ScienceFocus magazine is out now. Pick up a copy wherever you buy your favourite magazines or download us on your app store of choice. You can also find us on Apple News or online at sciencefocus.com. This podcast is sponsored by Name, Audio and Focal. The texture and emotional depth of music can be lost through digital sources or poor signal. Name audio believes you can have digital precision with analogue warmth.

Alongside French acoustic specialist vocal, Name creates high-end audio systems combining innovation with craftsmanship, so you can listen to music, just as the artist intended. Discover more at name audio.com.