The mathematics of creativity - with Marcus du Sautoy

When we're at school, we often asked to

make a choice. Is it Shakespeare or the

second law of thermodynamics?

Is it debusy or DNA?

Is it Reubins or relativity? Art or

science? which direction are you going

to go in? When I went up to my local

comprehensive school, uh the age of 11,

I was kind of frustrated by this demand

of the education system to ask me to

make a choice between these two. Uh it

was the first time I'd been in a in a

lab, the scientific lab, and I was

terribly excited by the power of science

to tell us um where we've come from, a

big bang, where we're going towards. Uh

so I fell in love with the world of

science to to give us an idea of our

place in the universe. But at the same

time I was also got the opportunity to

learn the trumpet. I played in our local

uh youth orchestra in Oxford. Um I

enjoyed uh signing up for the theater

that our school did. Um I enjoyed the

creative arts as well. And so I I found

it deeply frustrating uh when as school

time went on I was sort of asked to make

a decision between these two. But I was

very lucky uh my uh school to have a

maths teacher uh Mr. Balson who when I

was about 12 or 13 uh in the middle of

the class said dotoy I want to see you

after the class. And I thought oh gosh

I'm in trouble now. And um he took me

around the back of back of the maths

block and I thought oh I'm really in

trouble now.

Um uh but then he said doto I think you

should find out what mathematics is

really about because it isn't what we're

doing in the classroom all of this long

division signs and cosiness and things

like that it's something much more

beautiful and he recommended a few books

to me which

uh he thought would give me some idea of

what what maths was was really about and

uh I went up uh to Oxford that weekend

and we went to uh a bookshop and I got

these books and it was being like given

a key to a secret garden and these books

just opened up just the beauty of

mathematics and uh one book in

particular which made a very big

impression on me um was called a

mathematician's apology. It's a book by

GH Hardy a a number theorist. Um but in

this book he describes not exactly

mathematics but what it means to be a

mathematician.

And I began to understand that being a

mathematician actually could be a way to

bridge uh these two divides between

science and art because Hardy described

being a mathematician uh in in such a

creative you know he said a

mathematician is a creative person. A

mathematician like a painter or a poet

is a maker of patterns. I'm only

interested in mathematics as a creative

art. And he gave uh a few proofs of some

mathematics there which felt like uh

being told some fable or some story. And

I I began to realize that mathematics um

is a wonderful place where you can be

truly creative. So I decided that um I

would not have to make a choice because

I could be a a mathematician.

Mathematics is clearly the language of

science. And yet somehow it had this

connection to the creative arts as well

being compared to uh being a painter or

a poet. So I chose the mathematical

route. But in the years that I've been a

mathematician, I've spent uh a lot of

time still indulging my passion for

music, for theater, uh for the visual

arts, architecture, and I spent a lot of

time uh had the the privilege of being

invited into um the artist studio or the

um theater company's rehearsal space um

or the concert hall to work with

orchestras um to explore a little bit

about what they do And a as I've spent

this time with creative artists, I've

realized not only is mathematics

actually much more creative than people

realize, actually the arts themselves

have a lot more structure and

mathematics in them than I think many

people realize. And uh I like this quote

of Stravinsky's who says you know if you

talk to a creative artist structure is

incredibly important to what they do.

Stravinsky wrote I can only be creative

under huge constraints. Just given the

the blank stave or the blank stage um or

the blank canvas uh it's very hard to be

creative with nothing uh to give you any

sort of structure. So artists very often

enjoy choosing some sort of structure to

help their creativity.

Now for me if I was going to define

mathematics

uh it's not about arithmetic and long

division uh you know I think a lot of

people think what is it this guy is

doing in his office here all day in in

Oxford is it long division to lots of

decimal places and uh you know surely a

computer has put me out of a job but

actually for me I would define

mathematics as a study of structure

and that is the connection that art

depends on a lot of structure

for its creativity. And Stravinsky kind

of recognized this. He was aware of the

power of mathematics to be able to

stimulate his creativity. He wrote the

musician should find in mathematics a

study as useful to him as the learning

of another language is to a poet.

Mathematics swims seductively just below

the surface. But a lot of other artists

actually don't realize that they are

exploring mathematical ideas. Sometimes

they are discovering mathematical

structures for the very first time

before the mathematician or scientist

realizes that they're important

structure for understanding the

universe. Um so I've uh decided to sort

of pull all of these stories that I've

gathered over my years um spending time

in with theater companies like

Complicite or um uh uh musicians,

composers. So I brought all of these

stories together um into a book which

sort of explores how much mathematics

there is bubbling underneath the

creative art but similarly also to

demonstrate within the book um how much

creativity is important for discovering

mathematics and the two really feed each

other. And I structured the book um as

um nine different mathematical

structures uh and exploring how artists

have used these particular mathematical

structures um in their work. I I've

called these structures blueprints

because I feel like they are blueprints

uh for the way that an artist might sort

of fill out these structures. And what I

want to do in this talk is to give you a

little taste of three of these

structures and the way that artists have

used them for their creative um ideas.

>> The three things you're going to do are

going to go better if we

>> Yes, absolutely.

just interrupt you.

>> Yes. Okay. Oh, that sounds better. Yeah,

it was disturbing me as well. It's kind

of clicking. Um but uh great. So, I'm

going what I want to do is to share with

you um three of the structures. And I

think one of the interesting things for

me when I was writing this book is you

know I actually had to make a decision

about how I structured the book. Um I

did think that uh originally I was going

to start with artists and let the

mathematics kind of bleed out of the

artists. Um uh um because you know

people are frightened a little bit of

mathematics but I actually was a little

bit braver and I decided you know I'm

going to start with the maths and let

the artists um then explore the

mathematical structures. And it it

worked quite well because one of the

fascinating things for me writing the

book was to take a structure and just

see how differently it's interpreted in

say music or architecture or visual art

or literature. Um so uh it's kind of

curious that one structure and that's

why I feel that blueprints is a kind of

good name for them can be realized in so

many different ways. Um now, uh I if I

was going to give a blueprint blueprint

for the whole book, it would probably be

a triangle. And we've got two of the

subjects uh that are at the corners of

these triangles. We got mathematics in

one corner, the creative arts in the

other. But these structures that both

the mathematician is interested in and

the artists are interested in very often

uh you can already find them in the

natural world.

Those structures are actually how the

universe is built. and and sort of gives

us some understanding about why

mathematicians and artists might be um

interested in discovering and

investigating the same structures

because we're all interested in our

place in the natural world in the

universe. Mathematics an amazing subject

to explain the universe but artists too

are reacting to the natural world around

them um and our place in the natural

world and very often their art is

informed by their place in in nature. So

we've got this equilateral triangle with

the three uh mathematics, the creative

arts and nature at its three corners.

Now I'm going to take you to the first

um blueprint which is one of my favorite

because it's actually an area of

mathematics um that is part of my

research and it's a slightly unusual one

in some ways. Uh you'll probably be

expecting me to talk about golden ratios

and Fibonacci numbers. They are

certainly in this book. Um but I'm going

to take you to a different set of

numbers to start with and this is the

prime numbers. Prime numbers those

indivisible numbers like seven and 17.

And prime numbers you could even say are

the blueprints for the whole of

mathematics because out of primes we get

numbers. Out of numbers we get

mathematics. Out of mathematics I'd say

we get the universe. So these are the

blueprints for everything.

And each book uh each chapter I've

started um with a story about how an

artist has used this particular

blueprint or mathematical idea in their

work. And for the primes I've started

with one of my favorite uh composers um

which is Olivier Messian. Uh Olivia

Messian I first encountered actually uh

when I was playing the trumpet in my

local youth orchestra in Oxford one

weekend uh with conductor from the

London Symph Symphonetta um came to

visit us and brought the score for

Tarangala. This is an amazing orgasmic

symphony with incredible trumpet parts.

Um I think my trumpet playing was at its

peak that day and and it had to be

because uh the the piece is absolutely

extraordinary. I fell in love with

Messier and started to explore more and

more um the the the pieces that he'd

written. And there was one piece that I

got from my local library um which uh I

I really fell in love with. But it was

only much later that I realized that

underneath this piece is an incredible

piece of mathematics at work. Um this

piece is probably one of the most famous

pieces of the 20th century. Um the

quartet for the end of time partly

because of the story. It was composed

whilst Messier was a prisoner of war in

Stellag 8A. Um in the second world war

he found three other musicians. Uh there

was a rickety upright piano which he

played. Uh there was a clarinetist, a

violinist and a chist. And he wrote this

piece uh the quartet for the end of time

which was performed for all the other

prisoners in Stellag 8A on the 15th of

January 1941. Um the musicians wrote

about how absolutely freezing this

performance was. the clarinetist's

fingers were sticking to the metal of

the clarinet. Um uh the piece starts uh

with bird themes played by the violin um

and the clarinet and Messian was uh very

interested in bird themes in nature. But

it's in the piano part where we find an

extraordinary piece of mathematics at

work. So the piano part uh which I've uh

got the score up here for you to have a

look at. Um the rhythm sequence is

incredibly patented. It's a 17 note

rhythm sequence which uh starts with

crotchit crotchet crotchet. Then goes

into a nice syncopated rhythm. Then it

ends with crotchet minim. And then I put

the red line in the place where the

rhythm stops and then it just repeats

itself. Just repeats the same rhythm

over and over again. So you think, well

that's kind of boring, isn't it? Um but

no, because Messian does something

different with the harmonic sequence. So

the harmonic sequence is a sequence of

29 chords that he plays on the piano.

And when the 29 chords finish, and I put

a red line where the 29 chords finish,

they just repeat themselves. So it's

just 29 chords and then same 29 chords,

same 29 chords also could be rather

dull. But when you mix the two, what

happens when the 17 note rhythm sequence

has finished? The chords are still

working their way through the 29 chords.

When the 29 chords finish, it's halfway

through the rhythm. Um, and so the the

choice of 17 and 29 is very clever

because you'll see that they're both

prime numbers. And the 17 and 29 keep

the two things, the rhythm and the

harmony out of sync such you never hear

the thing come back to the beginning

again. It needs 17 times 29 chords by

which time the movement is already

finished. Um uh so uh just imagine if he

chosen 15 and 30 that would repeat

immediately 15 15 and already you're

back inside in sync. So the choice of

prime numbers creates this incredibly

unsettling effect of nothing quite

repeating although you feel like

something is repeating underneath this.

So I thought we would just listen to um

the opening of the quartet for the end

of time. You can hear these two primes

beginning to to work um against each

other.

So that's the rhythm sequence beginning

again, but the harmony still working its

way through those 29 chords.

And now the chords are finished and when

they're starting to play again, it's a

completely different rhythm. So, it's a

wonderful effect. And I was kind of

intrigued. Did Messianne know a lot of

mathematics to be able to do this? Um,

he's actually a character that comes up

quite often in this book because there

is so much mathematics bubbling um in

his music. And I asked George Benjamin,

a composer who's worked with Messianne,

um, how much mathematics did he know? He

said, Messianne didn't know any

mathematics. He came to these structures

for their aesthetic value musically. So,

it's almost like he discovered the prime

numbers um through his investigation of

the way that they can work to keep

things out of sync. Th this is a trick

actually that is used not just by um you

know wonderful classical musicians like

Messianne but also radio head for

example loves disrupting rhythms. Um so

most pop music is just in fours eight

16s rather boring. Um but radio head the

reason that I love radio head is that

they also like to disrupt rhythms using

different kind of prime number beats. Um

um so this is everything in its right

place which is almost the opposite of

what's happening in the music.

If you try and count what's happening,

you you find there's five beats in the

bar and then it goes to six and then

down to four. You never quite know where

you are. You try and dance to this, it's

almost impossible.

>> And I was very intrigued to discover

that the person who's responsible for

most of the fantastic composing in Radio

Head is Johnny Greenwood. Johnny

Greenwood when he was playing in his

local county youth orchestra when he was

a kid also played some Messiah wasn't in

my orchestra and he fell in love with me

himself because of being exposed to it

and I think you know understood the way

that you can use interesting numbers to

create effects um in the music. Um so

the fascinating thing as I said is often

these structures are already in the

natural world and there's a beautiful

example where nature was doing this

thing of keeping things out of syncs

using prime numbers way before radiohead

and Messianne came on the scene. There's

an amazing cicarda. It's kind of like uh

my desert island insect. If I had to

take an insect with me, it would be this

cicarda which has this incredible life

cycle. hides underground doing

absolutely nothing for 17 years. Then

after 17 years it emerges into the

forest. They party away. Half of them

sing the males to the females. Uh and

then after 6 weeks the uh cicardas all

die and the forest goes quiet again for

another 17 years. Absolutely bizarre

life cycle. How does it count to 17?

That's also not clear. But why 17? Is

that just a coincidence? It's the same

prime that Messiah used. Well, we think

not. There's also a species of sicarda

which uses a 13-year life cycle, but

there are none that use a 12, 14, 15,

16, or 18. So there must be something

about the primes which is helping this

sicarda. And we think it's the same

thing that happened with the messia.

Imagine a predator that appears

periodically in the forest as well. So

we're going to make the predator appear

every six years. What if the cicarda

appeared every 9 years? Well, they're

going to get quickly in sync because the

second time the cicarda appears, year

18, it's divisible also by six. The

cicarda gets wiped out. But now change

the cicarda to appear more often in the

forest every seven years. But because

seven is prime, it keeps out of sync of

the six-year predator, and they won't

meet until year 42. That prime is

helping the cigard to survive. The

predator dies out. Didn't know its

maths. Dies out. Good message in this

world. you know your maths, you survive.

Um, and so it's the same idea. The

predator and the cicarda are like the

rhythm and the harmony in Messiah. The

primes are being used for the

evolutionary survival of this cicarda.

And this is what's so lovely. A time and

again with these structures that I see

in the book, uh, often they're already

there in the natural world. So we've

seen a wonderful piece of music, two

pieces of music that use the primes. Um,

where else do they get used? Well, one

really curious place that I discovered

that primes are really important is in

the work of William Shakespeare. Now, I

was quite surprised because William

Shakespeare I generally regard as a

wordssmith, not not as a um a nerd of

numbers. Um and actually I got asked by

a festival to um do a presentation about

Shakespeare and maths for um one of

Shakespeare's anniversaries. And I I was

like a bit stumped. I didn't know what

to talk about. Um uh but the lovely

thing about uh being a fellow in an

Oxford college is that I spend a lot of

time talking to people from other

disciplines. And one lunchtime I got to

sit next to Will P who's our Shakespeare

edit expert and I asked him look I got

to give a talk about Shakespeare and

math I don't know anything about this

and he said no actually there's a huge

amount of mathematics bubbling

underneath Shakespeare's work. Um you of

course all know that Shakespeare writes

in aamic pentameter. Um, so there's

already a little bit of pattern there.

10 syllables, five groups of two, short,

long short long short long short

long short long.

But he doesn't always do that. When he

wants to disrupt things or wake you up

out of your kind of uh, soporific amic

pentameter, he mixes things up using

prime numbers. What's the most famous

line of Shakespeare? To be or not to be,

that is the question. To be or not to

be, that is the question. It's 11

syllables. So, you're sitting there

listening to your Hamlet and then

suddenly this thing jolts you. It's 11

is a very indivisible number. Doesn't

mesh with anything else. And suddenly

you're jumped into listening to the

speech. Uh another um line from McBth um

is this a dagger that I see before me?

It's also 11 syllables. And he didn't

need the that. He could have just missed

out the that and it would have been 10

syllables. But 11 is his way of saying

this is important. And if there is magic

a foot in Shakespeare, which often there

is, this goes down to seven syllables.

Um, so Mr. Night's Dream when Puck is

squeezing in the the love potion to make

the lovers fall in love with the wrong

people. Um, what does he say? Ch upon

thy eyes I throw. All the power this

charmed. O, it's seven syllables. Seven

was his code for there is magic

happening now. When shall we three meet

again? The witches also speak in sevens.

So Shakespeare was using these prime

numbers like a code to tell you

something. And of course if you look at

Shakespeare's time unlike when I went up

to school where this divide seemed to be

forced on you. Mathematicians and

artists and theater makers and musicians

were all talking to each other. John D

is producing the first uh volume uh

printed volume of the Uklitz elements.

It's quite likely that John D and

Shakespeare met each other. So prime

numbers can be used in many different

ways both in literature and in music um

and elsewhere as well. All right. So

that was a kind of number structure. I

want to take you now to a visual

structure, a geometric structure because

um they could be quite clearly used

maybe by say a visual artist. Um and I'm

going to take you to the idea of the way

that a fractal can be used by a creative

artist. Um the story that I start uh

this chapter with this blueprint is a

story of an artist that um also didn't

know what he was doing mathematically

but created in his artwork an amazing

examples of these geometric structures

called fractalss. So a fractal has the

property it's a geometric shape that if

you zoom in on a fractal it never seems

to simplify. it sort of has infinite

complexity. A a as you try and simplify

the structure, it never does. And so the

artist who tapped into this idea for his

um creative work um is Jackson Pollock.

So Jackson Pollock is an interesting

artist. Um partly his his paintings have

sold for some of the um you know

record-breaking prices at auction. But

the quality of Apollo um that's kind of

interesting um is that what he is making

has a fractal geometric structure. Now

uh I think a lot of people would sort of

think well gosh couldn't anyone make a

pollock you know I mean all he's doing

is flicking paint around. Uh I mean when

my kids are painting at home quite often

the um the room looks like a polock

after we've finished. Um uh so maybe you

could just fake a pollock. And indeed uh

uh I tell the story of somebody who

discovered 20 unknown pollocks in his

attic. Um and uh you know and they

looked incredibly convincing but it was

a mathematical analysis revealed that

actually these were fakes because

Pollock's painting have this wonderful

quality that as you zoom in on them they

never seem to simplify. Obviously, at

some point you start to see the pixels

of paint, but here there's four uh

images from one Pollock and I've zoomed

in closer and closer and closer. You can

probably tell which is the closest one

because you're starting to see the

texture of the paint. But the other

three, I think it's quite hard to tell,

you know, which is the next nearest,

which is the uh the far picture. So

actually um if you go around um uh from

the top left round to the top right um

in an anticlockwise that is us zooming

in on this picture and I think that's

what makes a polock so exciting because

when you're in front of a polic you you

sort of lose yourself in the painting

you don't know whether you're near or

far it's sort of uh that sense of

scalelessness is what makes these

paintings I think so special. So why is

it so difficult then to fake what

pollock was doing? Why um doesn't

everyone able to produce uh this kind of

fractal geometry? Well, Pollock had a

very unique um painting style. Um uh if

we were going to fake a Pollock, what we

would do is um put the paintbrush in the

paint pot and then we would start

flicking like this. But look what I'm

doing. I'm actually making myself in

just a very simple pendulum. And

pendulums are incredibly regular. So the

the pattern that I would make by doing

this um will actually have a regularity

to it as well. But what Pollock was

doing was not just flicking like this.

He almost danced um his way through the

painting. So he would move his shoulder

back and forth. Um the whole of his arm

would be moving. He would be moving as

well. Um uh it's of I mean he was a big

drinker so maybe he was also drunk at

the same time. Um uh but this actually

uh meant that what he was doing was not

a simple pendulum but an example of a

double pendulum um which is something

that we know is chaotic in nature. A

double pendulum is very hard to predict

and is chaotic in nature and the

geometry of chaos is a fractal. So if

you create create a chaotic system, put

a paint at the end of it, uh set this

thing off, then what you will see on the

canvas, um is an example of a fractal.

Um so actually there is a way to fake a

pollock, which is you set up one of

these chaotic pendulums, you put a paint

pot at the bottom, and um here is my

attempt to fake a Jackson Pollock um uh

using a chaotic pendulum. I put this on

eBay and it didn't sell for anything at

all. very disappointing but I'm still

working. It was only number one. So um

uh now this idea uh um of the fractal

you see is again related to the natural

world because I I went to Pollock Studio

and um it's in the middle of the

countryside and as you arrive you're

surrounded by forest and I went there

during the wintertime and the the trees

had lost all of their leaves and you're

essentially surrounded by fantastic

natural examples of fractals. A tree is

a good example of a fractal because it

has a big trunk then it branches and

then the branches called smaller

branches. The branches become twigs and

smaller twigs. The structure of a tree

is this kind of replication of structure

at smaller and smaller scale. And you

see that all over uh the natural world.

Take broccoli for example. If you take a

photograph of broccoli or a cauliflower

and then take off one of its florites

and take a picture of that closeup, it

sort of looks like the original

structure. So nature quite likes

fractalss because they're they've got

easy algorithms which is just kind of

replicating a structure at smaller and

smaller scale but yet can produce quite

a complexity. I mean for example uh my

lungs are fractal in structure sort of

main branch smaller and smaller. This is

actually very um so in in some ways what

was happening was Pollock was going in

from the outside fractalss that he had

in the countryside and then he was

making these kind of abstract versions

of nature. He was reproducing the the

natural world by making these in the

drip paintings that he did. And we even

have this idea of a fractal dimension

which measures how fractal something is.

Um it's a dimension between one and two.

One is just a line. Two is the canvas

filled with paint. And somehow these are

almost in between just a line and the

whole canvas filled. And the polocks

that we love are the pollocks whose

dimension is closest to the dimension of

the fractalss we see in nature. We are

responding to these pollocks because

they are an incredibly beautiful

abstract representation of things that

we are comfortable with, namely the

natural world around us.

This has been capitalized on because if

you want to create something which looks

very natural then you can use some

fractals. Um so another visit I made was

to um Pixar studios on the west coast of

America. And uh you might think well

Pixar is a lot of animators. Surely it's

just full of artists. Half the people

that are working at Pixar are

mathematicians who are implementing the

ideas of fractals to be able to create

very realistic um natural environments.

Uh and at the time they just produced uh

the film up this beautiful film uh with

the grandpa and the kid um and the

balloons and the dog. Um but it's the

natural environments behind which are

incredible. The jungle scenes are all

created using the ideas of mathematical

fractals. Um and another place um where

we've seen this is in some more recent

movies. You see one of the things that

uh I mean that's a very flat animation

the 2D animation and fractalss the ones

we know a lot about are very to

two-dimensional and this is the mandle

set which is this incredible um uh sort

of structure very simple algorithm which

generates it. But as you dive into this

um u anyone who who went clubbing in the

'9s will recognize these images because

they were the ones we danced to. Um but

um the current movies they wanted

something which is a little bit more

three-dimensional not this flat kind of

um space. So there was a question asked,

is there a kind of a threedimensional

version of this mandle set? And it took

mathematicians quite a while to tease

out how to make a kind of

three-dimensional environment which had

this kind of beautiful property of kind

of uh um almost like uh coral or or or

weird worlds going on. Um so uh we

discovered this thing a

three-dimensional kind of uh uh fractal

um which is called the the mandal bulb.

And it took some time to actually find a

formula which uh kind of teased this

out. Um but this structure when it was

shown to um uh the the guys who make the

the Marvel universe, they fell in love

with it and they decided to use it uh

the mandal bulb in in a few of their um

creations of weird worlds that you see

in the Marvel universe. One in

particular is one of my favorites. I

think of all the Marvel movies. My kids

are obsessed with them, so I have to

watch them. um Guardians of the Galaxy

uses this mandalbulb structure um to

create this thing called the living

planet in in the the second Guardians of

the Galaxy. Um so these are very

powerful tools for creating uh natural

and rather um weird alien uh

environments. Um now I want to come to

my third structure and uh this is uh

again a geometric one. Um, and what

we're going to do is I want to give you

a a beautiful case study and and really

not only just talk about but to really

show you how a mathematical structure

gets used by a creative artist um to

make something in this case a piece of

music. Um so we're going to explore uh

and we're going to play for you a little

bit later on um a piece of music uh

which is uses a platonic solid um for

its creation. So the platonic solids are

things like um this uh docahedron

uh tetrahedrin. Actually artists have

been obsessed with these kind of shapes

uh for a long time. You find um uh them

being drawn by Leonardo for example. Um

actually there some of the first uh

creative acts of uh the human species

are taking stone balls and carving um

shapes which are basically arranged in

platonic size. These are 5,000 years old

um stones that we've discovered uh

Neolithic stones um in Scotland. So uh

again we see the artists kind of

discovering these platonic solids before

the ancient Greeks started to discover

them. So, we know there are five

platonic solids. The one I'm going to

talk about is probably the simplest and

the one you know uh very well, which is

the cube. Uh the cube is an example.

Oops. You know, my cube is falling to

pieces. Um so, I'm going to I want to

talk about a piece of music that uses

the cube for its construction. This is a

piece of music um by uh the uh Greek

composer Janistaris, 20th century

composer. Uh he's kind of interesting

because he was also an architect and he

came from Greece um after the civil war

there. Uh he was in Paris and Lucabuzier

um took him on as part of his studio and

and Lucabuzier and Zanarchis actually

worked a lot together um creating um

some of the some buildings. The pavilion

uh the Brussels pavilion is one that you

know Zanarchis um was quite responsible

for. Um uh Lucabuzier is kind of

interesting. He's in my book because he

was somebody who used uh Fibonacci

numbers for example for constructing his

architecture.

Um now I'm going to talk about one

particular piece of music um which I I

have become obsessed with. So I I I hope

you're going to indulge me because uh

the the the rest of my talk is is

basically um indulging my obsession with

this piece of music. Um uh it's uh

partly because I uh it's a p piece of

solo cello and I started learning the

cello about 10 years ago. So uh I can't

play this and we have a wonderful chist

who's going to be uh performing it for

us. Um it's a po piece called Nomus

Alpha. It's a rather extraordinary

piece. Um and it uses the symmetries of

a cube for its construction. And what I

want to do is try and explain to you how

the cube is used for its construction.

He dedicates the piece uh interestingly

to three mathematicians

um including Everice Galwis um who was

one of the inventors of the language

which helps us to understand symmetry

called group theory. So my other

obsession as well as prime numbers is

group theory and this is a language of

Galwis that I use every day as a

practicing mathematician. Um so uh this

is anarchist very definitely is somebody

who knows about mathematics. Now um the

way this piece is structured there are

24 movements in this piece and what uh

Zanarchist does is to put musical ideas

that the cello can play on the corners

of the cube. So there are lots of

different things and you'll hear them in

the piece. There's the idea of piticato

where you pluck the strings or gissando

where you um glide up the strings.

There's another thing where you can turn

the bow upside down and hit with the

wooden side. So each of these uh kind of

eight different textures of the the

violin can play are put on the corners

um of the cube and we call them um S1 up

to S8. Now there are actually three

cubes at work um in this piece. There's

another cube which takes track of how

much time you're going to spend on each

of these corners and what the dynamics

are going to be. Because what um

Zanarchist does is in the first movement

he says okay um I put these musical

ideas on the corners of the cube and

then uh he composes a piece where he

traces out a path through the eight

corners of the cube. Now the path is

kind of interesting because uh you can

fit two tetrahedrin inside a cube. Uh so

a tetrahedrin is a uh shape with four

equilateral kind triangles um four

corners and so uh you can see one of the

tetrahedrrons here. So um the path that

you move through the corners um goes

from 1 to 2 to 3 to the four which maps

out the first tetrahedrin and five six

seven eight you get the second

tetrahedrin. So um this will be the

order that you'll play these musical

ideas and then there'll be another cube

saying how long you spend on each of

them and another cube which is saying

what the dynamic is for that particular

corner. Um and then he does a symmetry

of the cube. So for example, one

symmetry might be um there's an axis

running through uh the cube through the

opposite corners and you can rotate by a

third of a turn and the cube comes back

to where it was before. But now you see

these two corners have stayed the same

but the other corners have all moved

around. So after you've done the

symmetry, the textures have all been

moved around and so the next movement

will give will be played in a different

uh this the same path mapped out but the

textures will be played in a different

order. Same for the amount of time and

the dynamics. Um so for example so you

you start the cube off in the first

place and then he he does a symmetry and

then the piece starts with where the the

things are located at that point. So

here we see actually um his little

notation uh it turns out that the order

that things will be played in is given

by this first rotation which is uh

through the diagonal opposite here and

then each variation after that

corresponds to another symmetry of the

cube. Now I told you there are 24

movements and there are 24 symmetries of

a cube. um there I mean we can do you

know for example those diagonals or we

can do um through a face the axis

through a face or um opposite edges we

can also rotate there um if you count

them up there are 23 things you can do

um but we also include a another

symmetry which is leaving the thing

where it is kind of like a zero symmetry

so 24 symmetries so my initial

impression was great there's a movement

for every symmetry but as I dug into the

piece it turns out to be way more

interesting so the piece is actually

divided into six groups of four

movements. Three of the movements will

correspond to symmetries, but the fourth

movement is some sort of free form uh

flowing uh uh kind of morphing of the

cube. So I was like that's really

interesting. So there are actually only

18 symmetries. So how did he choose

which the 18 there were? And as I dug

deeper I realized he's was using a

really cunning idea. He starts with two

seed symmetries. So the first two

movements are u the symmetry with uh

through the first one is the symmetry

through opposite corners. Uh the second

symmetry is um corresponds to uh

rotation around uh an axis through the

middle of two opposite edges. But then

the third symmetry how does he choose

that? He combines the first symmetry,

does that with a cube, and then he

combines the second symmetry, and then

the combination of those is actually a

third symmetry that we could do in one

go. Um, so actually the combination of

those two symmetries as as if the cube

had just been rotated through a face. So

he gradually builds up each symmetry is

built out of the two previous

symmetries. You just combine those and

you get a new symmetry. Um, this

actually might be rather familiar

because it's the way the Fibonacci

numbers are defined. You add the two

previous numbers to get the next number

in the sequence, but Zarcus is doing

this with symmetries. And as I've

plotted out the the symmetries that you

get, um, you see this incredible path.

So sometimes you'll actually see the

same symmetry twice. Uh, the longest

path that you can make is 18 symmetries

before the thing repeats itself.

If you choose two other seed symmetries,

you can get back very quickly in six

symmetries. And I'm very intrigued and I

still haven't found out how did

Zinarchist find that 18 is the longest

that you can do in in when you take the

cube. Was it just experimenting? How did

he know it was going to be the longest?

So here is actually where the path um of

this uh um through the 18 symmetries and

some of them appear twice because you

can get them in different ways um uh

through this path. Now we have something

called the kaly graph of a um of a the

symmetries of an object. So the 24

symmetries are here. This is like a

little map. Now this little map is going

to be on the screen for you as we play

this piece because what I'm going to do

is to ask my chist in a moment to come

up and actually we're not going to do it

as a solo cello piece. We're going to do

it as a duet. I'm going to be on the

visuals up here giving you a guide to

the way the mathematics of the cube is

working as the piece is played. It's I

must admit it is I mean maybe it's I

shouldn't say this before you've heard

the piece. I think it is quite a

challenging piece to hear for the first

time. It's incredibly acrobatic. Um and

so what I wanted to do was to give uh

some visual guide to what is happening

as you listen to this piece. Um, so what

you're going to see is an animation that

I made with Simon Russell, um, who I've

been working with with kind of trying to

realize mathematical structures in

music. And what you're going to see is,

um, as the piece is played, you're going

to see the path. Um, actually I can show

you just in this. So there was one of

the symmetries and now the first corner

is being played. And as you hear the

textures change, um, this little ball

will run across the, um, the cube

playing different textures as you'll

hear them. Um, and you'll see on the

side there's also, uh, the symmetry

we're on and what two symmetries it's

been built out of. And we also have the

little kaly graph, which is the map of

where we are at the moment um, in the

the 24 symmetries that make up uh, this

group. So, this hopefully is going to

give you a little bit of a guide to how

Zanarchis uh kind of made this um piece.

I've actually done this with a few

chists who've come to this piece for the

first time and found this uh animation

actually very useful in learning the

piece. But um uh we're very lucky to

have uh tonight Alia Uova

who is an uh

um who is an expert in this piece and

knows it very intimately. Um and so I'd

like you to give a big round of applause

to our chist who's going to be playing

uh the piece.

Now, as I said, I'm going to be also

playing the piece. Um, but I'm going to

be playing the visuals. Um, so I also

have to follow the the score uh like a

hawk. Um, so uh I'm just going to bring

up the Is that good? Right. Uh,

fantastic.

So there there's our cube and and and

it's ready for the um first Now I need

my glasses for this as well.

So this is Nomous Alpha by Janice

Zanakis duet for cello and VJ

Mhm.

Let's

go.

Oops.

Come on.

Heat. Heat.

Stop.

Stop.

Heat. Heat.

Heat.

Heat.

Thank Thank you guys.

Uh so thank you for that. The symmetries

of the cube made into music. Uh but for

me one of the exciting things about this

dialogue between the two the creative

arts and mathematics is that you know to

do mathematics you need an incredibly

creative mindset and by working with the

creative artists that I have over the

years I think it has um stimulated the

ideas that I've developed um

mathematically. So this is really not

just a one-way uh conversation. uh it's

not just the musicians or the uh visual

artists plundering the cabinet of

wonders of the mathematician. Very often

the artists are discovering these

structures for the first time. And I

think what's especially interesting

about the time I spent with this piece

is that um that idea of a Fibonacci

sequence of symmetries. Uh I've seen it

in the Fibonacci numbers. Sure, it's

very well known. But the idea of taking

two symmetries, combining them together

a and creating another symmetry. Um

unlike the Fibonacci numbers which go

off to infinity, if you take any um

symmetrical object, um these do

eventually come back and repeat

themselves. And one of the questions is

if you take two symmetries, um what is

the longest path for any symmetrical

object? The cube it's 18 that you can

create. But what about a docahedron or a

tetrahedrin? and and the time that I

spent with this piece of Zanaris

actually has resulted in me going in a

new direction mathematically. Um and

that's I think so beautiful the example

of spending time with a piece it asking

new questions um which now uh the thing

that I'm spending a lot of my time doing

can you work out is there a formula

which take a number a shape with a

particular number of symmetries what

will that longest path be and it seems

to be a a very wild problem that is

asking about symmetrical shapes so I

think that's why it's so important that

we maintain this dialogue and don't ask

people to separate are Are you a

scientist? Are you an artist? Every

scientist is an artist and every artist

has science and mathematics bubbling

under what they did. Just as Stravinsky

said, for the creative artists,

mathematics is like another language is

to a poet. It bubbles seductively just

below the surface.

Thank you very much.