Tiger. Tiger burning bright in the forests of the night.
What immortal handle I could frame the night before symmetry.
As humans, we seem to be incredibly sensitive to the idea of symmetry.
When we were in the jungle and we wanted to survive.
It was those that could spot symmetry that had a chance to realise that there was a tiger out there.
If there's something with symmetry, it's likely to be an animal and either it's going to eat you or perhaps you could eat it.
So nature has kind of programmed us to be incredibly sensitive to notice things with symmetry because they've got a message inside them.
Very often they're things that we should notice. Certainly many animals are very sensitive to symmetry if you're in the garden.
A bumblebee looks out for things with symmetry because it knows that that is likely to be sustenance.
The flower, in its turn needs the bee, and so it needs to draw the bee to its flower.
And so symmetry is acting a little bit like a language communicating between these two, the bee and the flower.
And for us, too, as humans, we use symmetry to communicate information.
I've got two theses here which I'm going to make artificially symmetrical, and I ask you, which of these do you find more beautiful?
The first two or the second two that I've made artificially symmetrical and most people are drawn to the second two.
Why is it that humans find symmetry and a human face so beautiful?
Well, again, this is communicating information. It's hard to make symmetry in nature.
So this is an indication that this person probably has a good genetic background, has been brought up well, is healthy, is going to make a good mate.
So the symmetry is communicating to you information about the person that they probably be good to have children with.
And Galileo certainly highlighted the fact that symmetry is a way of reading the world, of understanding the universe.
He very famously wrote, The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written.
It is written in mathematical language, and the letters are triangles, circles and other geometrical figures without,
which means that it's humanly impossible to comprehend a single word.
Certainly, symmetry is very central to how we understand the scientific world, but it's also be very important to artists, too.
And many artists have talked about the the role of symmetry in their work.
This is Pushkin,
who wrote in a letter that symmetry is a characteristic of the human mind and it seems to be that we've been evolved to recognise symmetry.
Valerie French poets wrote The universe is built on a plan, the profound symmetry of which is somehow present in the inner structure of the intellect.
And it's perhaps intriguing that both of these were drawn to a literary form that has a lot of symmetry at work in the creation of poems.
But it's not always such a happy relationship.
Many artists find symmetry a little bit too controlling and a little bit of an uneasy relationship.
Here's Thomas Mann talking about symmetry in the Magic Mountain.
He has a character describing the snowflakes that land on his arm.
The character said he shuddered at its perfect position, found it deftly the very, very marrow of death.
But for us, as mathematicians, symmetry is far from deadly.
It isn't still. It's something very active and something with a lot of energy and movement in it.
And the person who identified that symmetry is something with energy in it in some ways was a very romantic figure in the history of our subject.
Evariste Galois, killed in a duel at the age of 20 over love and politics.
But already when he was still in school at the age of 18,
he'd come up with a new way of looking at symmetry, which revealed that it's full of motion and movement.
So if you take a wall, for example, in the Alhambra and you ask, what are the symmetries of this wall for Galois and for mathematicians,
this is about the movements that you can do to this figure, which makes it look like it was before you moved it.
So we trace an outline on the this particular wall in the Alhambra.
We can move these tiles by 90 degrees and then they fit perfectly back on top of each other.
And this for someone like Galois, is the element of symmetry.
The ways you can move something, transform it, such it has a connection to where it came from, but it's evolved into something new.
And this is an idea that one particular art form uses a lot to create an idea of movement, and that is music in.
Music very often you starting with a theme and you want to somehow mutated and change it into something else,
which has a connection to where it's come from.
And I think probably the musician who captured most the idea of using symmetry to generate interesting new ideas is Bach.
Metzler Bach's student once described Bach's music as the process of sounding mathematics.
Now, as you came in, you were, in fact, listening to what I think is one of the pieces, which is almost like a hymn to symmetry.
It was the opening aria of the Goldberg Variations.
And for me, the Goldberg Variations is Bach really exploring the idea of symmetry in sound and music?
So the Goldberg Variations, a piece for solo piano, starts with this aria, which is what you were listening to when you came in.
And then Bach goes through 30 variations on this aria and then concludes with the same aria.
So already you have a sense of a circle that the thing joins up the beginning and the end are exactly the same.
And to really reinforce this idea of a circular structure happening inside the Goldberg Variations,
if you move to the 16th variation, which is halfway through the piece.
Bach intriguingly calls this an overture.
Now, an overture in music is usually something which begins the piece.
So already you're questioning where quite is the beginning of the Goldberg Variations.
Maybe you can start anywhere and join the whole thing up in this circle where Bach really plays with the idea of of symmetry to create an idea,
a variation on the theme that we've heard in the ARIA is in every third variation.
Every third variation in the Goldberg Variations is something called a canon, which you probably all remember from school.
A canon is where one voice starts singing.
And then a little bit later, the second voice comes in and starts singing the same thing, but with a delay in time.
And this creates a beautiful effect if you choose the right the right tune.
So every third variation is an example of a canon.
So here's the first canon, the third variation.
And so I'm going to play a very short snippet of this and you will hear the first voice starting.
It's a start on this B and does a nice little trill up and then you hear the second voice coming in doing exactly the same thing.
So let's hear the first canon, the Bach heads in the Goldberg Variations and listen out for just the repeat of the pattern.
This is like a pattern which is being repeated, like a tile across a wall.
There it is, then tilted.
So over the two bars you had to fill the first voice and the second ball came in with a second voice.
But Bach isn't content with each new canon repeating this idea with just a second voice repeating what the first voice did.
So he decides to take the pianist on a journey.
So in the second canon, the sixth variation, the second voice doesn't just repeat what the first voice does, but starts.
We've seen a shift in tone, the sort of tile moving this way.
Now we're going to hear a shift in pitch because the second voice starts one note higher in the third variation.
So it's little different out today. So you hear the the G cascading down and then you hear one step up in a casting Kafkaesque cascading down.
So each time you hear the first voice, but the second voice is one note higher.
So as each new canon is introduced,
the second voice climbs gradually higher and higher and higher until something quite interesting happens when you get to the eighth variation.
The eighth canon, the 24th variation. Because at this point, this the second voice has been stepping up one step at a time through the scale.
And by eight notes, it actually hit something called the octave.
We actually get to a note, which sounds like the note you started with, but just an octave higher.
So Bach kind of joins up this circle again.
In this case, actually, it isn't higher, but lower. So you'll hear the first voice start on a G and then an octave lower, in fact.
So he varies quite where this second voice is coming in.
But you suddenly hit this point where suddenly the two themes join up again as if they're starting on the same note, but just an octave higher.
So you heard the second boys come in there. But we still got one more cannon left.
So where does this voice go on then, when in fact, it takes that next step upwards?
So for me, when I hear this, these cannons and what I'm hearing, it's actually a kind of geometric shape embedded in here,
because we've already heard the circle where we've had the ARIA starting and it joins up again with the ARIA at the end.
But then these cannons are creating a second worth of circles in this piece, or maybe even a spiral because you're getting this octave effect.
So embedded in this piece is somehow the shape of this Taurus, a circle's worth of circles.
And as I said, the ninth variation continues this step upwards as if you it's telling you where the piece is going to go,
if you want it to it, extend it off to infinity. And indeed, the first voice comes in.
The second voice is an octave and a note. Nine notes higher.
Now, Bach also uses ideas of symmetry to do these variations.
So sometimes the second voice isn't a complete copy of the first voice, but actually has some symmetrical operation acting on it.
So for example, the first voice might shoot upwards and then you'll hear the second voice shooting
downwards as if there's been a reflection in the horizontal line through the music.
So here we see the note going.
It goes down and then up and then down. But the second going it goes up.
Then down, then up. And then. But also likes to use a bit of symmetry at work in the rhythm and structure of these canons as well.
So if we go to, for example, let's go back to the 24th canon.
He's broken up the beats in this bar into three beats, and each beat in turn is divided into three triplets.
So we get a kind of three, three, lots of three in this structure.
So let's hear this one at work again. One, two, three, and one, two, three.
One, two, three. One, two, three. One, two, three. One, two, three, one, two, three.
So Bach has some different combinations that he can do in all of these canons.
Here we see a ball divided up into three and each beat divided up into three.
But there are different possibilities. So we see Bach, the mathematician, at work who says, Well, what are the other things I could do?
So I could divide the bar up into four beats or two beats and each beat I could do into triplets?
Semi quavers or quavers. So he goes through all the different possibilities that you could have with these different ways of breaking up the rhythm.
So this is a bit like having a combination lock where you've got two triangles.
One triangle is controlling how many beats are in the bar? Two, three or four.
The other triangle is controlling how that beats is broken up into quavers, triplets or semi quavers.
Bach makes sure, as if he's trying to break this combination lock, that he covers every particular possibility.
So how many different ways are there to arrange these three. Two triangles on my little combination lock.
There are three times three nine different ways. We had nine canons that we saw already.
So indeed, Bach makes sure that each canon covers one of these different possibilities of the symmetries inside the different rhythm.
So we've seen three broken, three beats broken up into triplets.
For example, variation number 15, we see two beats broken up into semi quavers.
That's. And I think this really illustrates that BARC is not doing this kind of unknowingly.
I think that would be very hard to do this unknowingly. I think Blanck is very aware that he wants to make sure that this structure covers
all of the different possibilities and there is so much structure in here.
You've seen how the canons build up. They climb each time, the rhythms and things.
But we've got one more cannon that we haven't heard, which is the last variation, the 30th variation.
So you might think, well, I know exactly what is going to happen now. It's going to take a step upwards again, perhaps some start the rhythms again.
But no, when Bark comes to this last variation, he destroys the structure completely.
And this last variation is called a quarter limit, a musical joke.
It's in fact, just a variation on two folk themes of the day has nothing to do with the rest of the structure of the piece, which is kind of curious.
But I think it goes to illustrate how much structure has been up to that point
that you find this suddenly the two folk tunes woven together so surprising.
And this is not uncommon with artists. I think art is very often like this idea of setting up symmetrical expectations and then breaking them.
One of my favourite examples I work with a professor in Japan.
Professor Kurokawa. And we went to he took me up to Nikko and I took this photograph.
It's of an arch. As you enter the temples and on the arch there's eight columns.
There's a beautiful pattern, symmetrical pattern on these eight columns.
Seven of them are exactly the same, but the eighth one is turned upside down.
And I said to Professor Kurokawa, wow, the the architect must have been really angry,
the builders, that they got this wrong and turned it upside down. And he said, no, this is a very deliberate act.
And he referred me to this wonderful Japanese essays in Idleness from the 14th century, wonderful book title.
I need to write a book called That, where the essayist wrote in everything, uniformity is undesirable.
Leaving something incomplete makes it interesting and gives one the feeling that there is room for growth.
And I think that's what artists like doing. They like setting up this expectation and then breaking it.
And in the Goldberg Variations, that upside down column is that 30th variation when I guess a ball is probably one of the
the musicians who use symmetry a lot and it's a classical period in the modern period,
probably the person who took up the mantle of using symmetry to do themes and variations is Schoenberg.
Schoenberg system that he introduced has symmetry at its heart to generate a palette of themes that he would then compose with what he would do.
He broke the kind of harmonic structure of the major minor scale and said every note of the chromatic scale 12 note should be considered as equal,
and they will then do some sort of variation of those 12 notes and arrange them in some interesting way.
And then you would do symmetrical operations on these 12 notes.
So this would be called a 12 tone row, which you would then you would choose a 12 tone row that would be your kind of theme,
and then you would do these variations on this in a very mathematical way.
So the first variation is really to kind of push. If you think of this like a little tile, you would push the tile gradually higher and higher.
So each note would be taken one semitone higher and you might drop a one down if it goes
above the octave and you would generate 12 different shifts of this particular motif,
which all have pretty much the same shape, but it's just a different pitch.
So you've got 12 those. So those would be little translations, but then you would do some more symmetrical moves.
So you would do a reflection in the horizontal line, you would flip the thing upside down, just as Bach did.
Remember that with that theme and variation.
So Shambhu would say, You turn all the 12 upside down, but you can also vary them by playing them backwards.
So he would reverse them and do a reflection in the so that was the horizontal line in the vertical line.
And then he would combine those which would be a combination which would be like a rotation.
So then you would have 48 different themes from which you would then start your composition.
And this was Schoenberg's kind of the way that he changed music at the beginning of
the 20th century by using these symmetrical tools and composers that were followed,
Schoenberg system were drawn to interesting 12 tone rows that when you made these symmetrical moves, perhaps interesting coincidences happened.
One of my favourite is Olivier Messiaen.
Olivier Messiaen was very interested in the ideas of using mathematics quite intuitively, very often in his music.
He used he created some new scale structures based on symmetry.
But he also was very intrigued by this system of Schoenberg's to use symmetry to create themes and then bury them with these symmetry moves.
But I particularly like one piece that he wrote for solo piano called Ildefonso, too,
because in this particular piece he chooses a 12 tone row, which has huge mathematical significance for anybody who studies symmetry.
This 12 tone row, it's a little bit like taking.
So you can think of the notes a bit like 12 cards, a pack of 12 cards, and there's a way.
So the notes will be arranged from, say, the ice through to the the queen and those will be your chromatic notes.
But then you want to make some new variation, some, some permutation of those notes.
So what Marcion was intrigued by was a particular permutation where you take the top and the bottom cards and put them on this hand,
and then you would take the next top and bottom card and put them on top of this hand.
And you would keep on doing this until you cleared this pack and you had rearrange the cards on this side.
This is a particularly. A shuffle called the Mongolian Shuffle.
And he used this. He said, Well, that's interesting. I wonder what sort of music that will create.
And so this is the the Schoenberg 12 tone row that he uses in this piece, Ildefonso, too.
And actually he doesn't use the techniques that Schoenberg used, but he said, okay, well, I can repeat that shuffle so I can keep on it.
I can take that permutation, take it over to here, repeat the particular shuffle I've done,
and that would be the next 12 tone row that you would hear. So I think if I got a piece of you know, I think it's coming up.
So this was the this actually is the piece goes on. You hear the shuffle being applied again and again to these 12 notes.
This particular permutation for a mathematician is one that we recognise because one of the Galois started
this journey that we've been on for 150 years since he first introduced this new way to look at symmetry,
which culminated in us being able to produce what we call kind of a periodic table of symmetry.
It's called the Atlas of Finite Simple Groups. So I brought it along here.
And this thing here is like the periodic table that the chemists have in the labs across the road.
It lists all the kind of atoms of symmetry from which we can make all four symmetries.
So there are some very simple symmetries kind of shapes with the prime number of sides.
There are symmetries which fall into kind of patterns.
But then there are 26 very strange symmetrical objects that seem to have no pattern to them at all.
We call them the sporadic, simple groups, and some of the first to be discovered were by a mathematician called Matthew.
And he he discovered one which uses this permutation that Messiaen was using.
And it it creates this extraordinary symmetrical object.
It's not an object I can show you. The first dimension that you will see this represented in, I think is 11 dimensions.
So I can't show you this symmetrical object, but wonderfully messier has given away as a way to listen to it.
So here's the sound of an 11 dimensional, sporadic, simple group.
And then it goes off into something else. But but that was the first permutation that you were hearing of the set of 12 cards.
The 12 notes. Amazing. The mission was drawn to this for aesthetic reasons.
He had no idea that this had a connection to a significant object for us as mathematicians with extraordinary symmetrical implications.
Another composer that one of my favourites from the 20th century who was drawn to a symmetrical object that you can see was Yanis Xenakis.
He's a Greek composer who worked with Le Corbusier. He was an architect, but he was also fascinated in mathematics.
He wrote a piece called Nomos Alpha, which he actually dedicates to three mathematicians, Galois included in them.
And this is a piece for solo cello. And he uses, again, the idea of symmetry to do a variations on a particular theme.
So this is I think I got the yeah, so so I'm going to play you the piece.
A little excerpt from Nomos Alpha.
I want you to try and see what sort of symmetrical shape is conjured up in your mind's eye by this particular piece.
And then, you know.
You can open your eyes now. What did you see? Did you see a symmetrical shape?
Anybody? It's very asymmetrical, I think. In fact, that was in fact, a cube.
Although I find it quite hard to hear the cube and I. And it's a little bit unfair to only play you that amount of music because you only start
to hear the cube and its symmetries and work actually as the variations begin to build.
Because then you start to see some connection between this the, the, the piece of music you've just heard and as it's varied.
What's going on here is that Xenakis took the 24 rotations of the cube, and what he did was to put musical ideas on the corners of that cube.
So I think we've got them here. So so the eight corners on the cube.
So I actually had several cubes at work at the same time.
So you see here s12s8 so this would describe the quality of the sound.
So you heard lots of different things that the cello could do, a sort of pizzicato glissando, and these are described by these particular shapes.
And then there'll be another cube keeping track of how long was spent on those particular elements and also how loud or soft they were being played.
So he would read the corners of the cube and then compose with those restrictions, and then he would play apply a symmetry.
So one of the symmetries that we've seen here to the cube and then the next variation would read off in a different order,
the things that the cello would have to do. So you'd hear these, these things varying.
So by actually the end of the piece, you've heard a sequence of permutations at work rotation,
symmetries of the cube, and you hear begin to hear some sort of interconnection between them.
There are 24 variations. But what I discovered just recently is that he isn't going through all the rotations.
He's actually produced a new structure, which I've never seen at work.
And since there are some group theorists here, they might tell me that this has been studied,
but he actually uses a kind of Fibonacci rule to generate which symmetry is going to be used next in the next variation.
So he explored, he took two symmetries of the cube, and then the third symmetry would be used to be the combination of those two symmetries.
Then the fourth symmetry that would be used with the combination of the two previous ones.
And what he did was to find which are the two symmetries which have the longest chain before repeats itself.
So he found a chain of 18 kind of Fibonacci things, which then return to the beginning again.
So you had 18 variations and then he would pit between every third one another variation.
So to bring us up to 24. So this is one of the beautiful things I find sometimes with working with collaborations,
but across disciplines between art and science that out of this perhaps comes a new question is that
an interesting structure for a mathematician to look at the idea of a Fibonacci chain of symmetries,
which I'm not sure I've ever seen studied. So it's an arc is using the cube as a way of generating those variations.
Again, we see symmetry as a way of restricting the possibilities that he could do.
He could have just done eight factorial different ways of arranging those kind of sounds for the cello.
But no, the cube restricts and constrains his creativity.
And this is often why a composer will use mathematical structures in their work because they re too much freedom and you can't create.
This is Stravinsky talking about the importance for him of constraints coming from the things like mathematics.
My freedom consists in my moving within the narrow frame that I have assigned myself for each one of my undertakings.
I shall go even further. My freedom will be so much the greater, a more meaningful,
the more narrowly I limit my field of action, and the more I surround myself with obstacles.
So we've heard a lot about the way that composers have used symmetry as a way to create this framework within which to compose.
But there's also something interesting working the other way,
because we've also discovered that within sound we can discover symmetry at work and creating that sounds.
And this is what we've got set up here is a demonstration of how symmetry is embedded in just the nature of sound itself.
And this is an experiment that was done at the beginning of the 18th,
19th century by scientists in Scotney who discovered that symmetry is actually hiding in the sounds that we hear.
He became very famous. He was kind of the Brian Cox of his day going on tours around Europe, giving lectures about physics.
And he discovered that if you take a metal plate and you take a violin bow and you vibrate
the bow and you put sand or salt or pollen on top of the bow with careful vibrations,
you can get some interesting kind of symmetrical patterns appearing in it.
And. And he went around the courts of Europe. He actually gave up his position university.
He managed to make enough money doing this tour.
He went to Paris, where he demonstrated this for Napoleon, who thought it was so extraordinary, gave him 6000 francs for his demonstration.
So what we thought we would do is to give you a kind of insight into the way symmetry appears in the vibrations of a plane.
So we set up here on the front here an example of a flat in his experiment at work.
And what we've done is a slight variation because we've chosen we put four plates together.
So we're going to see how each plate will produce some symmetry and how these symmetries
then build up to make perhaps something a little bit like the the wall in the Alhambra.
So now you've been given these earplugs, which are not, because in case you find the lecture so boring, you wanted to go to sleep.
They are, in fact, for this moment here, because what what we're going to do now, Aaron,
the frontier is going to be operating and picking out particular frequencies of the vibrations of this plate.
And we'll see the different shapes that are emerging in the salt that I'll put on the plate.
So you can now put these in and I will put mine in because I'm the closest, probably good the only to it.
Aaron got his fancy ones. He's got like, I need to get some of those on this.
Okay. So we're going to put the salt on and then as the frequencies.
So what we've got underneath is amplifiers which are going to make the plates vibrate.
So. Maybe.
Fantastic. You can take them out now.
Yeah. Thank. Now.
I'm not saying I found that magical the first time I saw this actually at work.
And if you want, you can find it. This is this was a YouTube video just in case that didn't work.
But but he documented all of these extraordinary different shapes, some quite simple, but as the frequency gets larger.
So what we have here is a little speaker with a roll to connect you to the speaker, and the road is vibrating the plate.
And we see these extraordinary patterns at work. So what is going on here?
Well, this is very similar to this kind of two dimensional version, really,
of the discovery that Pythagoras made thousands of years ago that music and mathematics are intimately related.
He discovered that the notes that we find harmonic and nice combined together have mathematics hiding behind them.
And you can demonstrate this with a string.
So if you vibrate a string, then you can vibrate it twice as fast.
And we see a point in the middle which doesn't move, something we call the note.
So this point is not moving at all. This is a note which is an octave higher.
If we vibrate three times as fast, we see two points which don't move.
And this is a note which is one perfect fifth and this is the basis of musical harmony, is the combination of these second two notes at work.
But the interesting thing for our flattening plates is that what we're seeing here, the two points which don't move at all.
The what we're seeing with the sand stabilising is these the plate, the lines in the plate which are not vibrating at all.
They're the nos. They're a bit like these stable points here.
So when we see all of these patterns at particular frequencies, there's a complete shift and change.
We've picked out the particular frequencies. If we done it as a continuous frequency increasing,
you would have seen a complete shift of the pattern from one to another as we hit each resonant frequency.
So the patterns you are seeing are the places where the plate is not vibrating.
The challenge was, can you mathematically explain why you're seeing these and even predict for different shapes what sort of patterns will emerge?
What sort of frequencies will you hear? Those patterns.
So as I said, Napoleon heard this lecture, was so excited about it, he said, this man makes sound visible.
And it was at this time that at the Paris Academy, they used to like to set prizes.
Challenge is to try and stimulate the mathematicians at the time and they choose different sorts of challenges.
And the one that was chosen, I think, in 1808 was to explain the patterns that were appearing at these points,
highlighting the experiment had found these. But how can you predict what you will see in these plates?
The prize didn't get many entries. The first entry, the two people injured.
One was became a member of the academy and wasn't allowed to enter any more because he became a judge.
The other one was a certain Antoine, the blonde. And this was quite intriguing for the professors at the time because this student, Antoine the blue,
have been particularly bad students and then suddenly was performing extremely well and they were kind of intrigued,
well, how come the student is suddenly become incredibly good?
The solution that was submitted was not really worthy of the prize.
But then a second solution came in. At which point they decided to find out who this was.
It turned out not to be Antoine Loblaw, who had left the University of the École Polytechnique some time ago.
But it was in fact a woman, Sophie Germain, who was using Antoine de Bloom's name because she could study at the École Polytechnique,
and so she had assumed a man's name in order to be able to try and get access to the university.
Her third submission for this prize was under her own name, by which time she'd been outed,
and it was a worthy enough contribution to the understanding of these plates.
Now that she was awarded the prize, it was the first time that a woman had won this prize given by the Academy in Paris.
The solution was not complete, and it seems it might have had some mistakes in it.
But Sophie, I mean, it's one of the things that she contributed.
The other one is very relevant to this building here because she was one of the
people who made a significant contribution to the solution of Fermat's Last Theorem.
She was able to show particular equations of the form X to the p plus winds appliqués into the p do not have solutions,
but the P had to be a special p prime such that also two times p plus one was also prime.
These are now. So for example, five, two times five plus one is 11.
So that's a prime which with you with this method would work. These are now called the Sophie's.
You mean primes. But she also made a contribution to the first attempt to understand the mathematics behind these plates.
But it wasn't until the beginning of the 20th century when Valter Ritz finally came up with a way to solve what was going on,
to be able to make predictions. He worked out what the frequencies were,
and they comparing very well to the experimental frequencies at which these particular vibrations were appearing.
His work wasn't really recognised at the time. As I understand it, Russians took his work and realised that it was very powerful.
It now is, I think, the basis of the finite element method which is used by many people solving pdes.
Lord Raleigh kind of claimed that he'd stolen his ideas.
So I think it's it's right that we celebrate somebody who perhaps name isn't so much attached with these plates as it should be.
And he came up with this way of producing the particular vibrations that gave a prediction for the particular plates.
But this particular mathematics and the physics behind these plates isn't just relevant to understanding
the vibrations that HLAD was producing in that wonderful lecture that he gave across the corks.
In Europe, we now use the same sort of ideas across so many different disciplines that this has an incredibly rich heritage.
The ideas that came out of these plates.
So, for example, musical instruments, it's very important to understand the particular resonances that will happen in particular shapes.
Why is a Stradivarius violin so much more valuable than a Yamaha manufactured factory?
Manufactured violin? Some people believe it's the nature of the woods that's made that the violin is made from.
But one of the things which is particularly relevant is the shape, the perfection of the shape of that particular violin.
And you can analyse the particular craton, the patterns that appear in the violin box.
Of course, you know, you've just got a, a string vibrating. So you might say, you know what's so important?
Surely the string is the most important thing,
but it's these resonances that are produced in the violin which are actually the characteristic of the violin.
A comes from these these patterns that you see if you take a circular plate,
which is one that we analysed and it's actually mathematically the easy one to analyse,
what you discover is that the first frequencies where these patterns appear are actually very related to each other.
They're in the first frequency and the second frequency are in a 2 to 3 relationship,
which is the perfect fifth from which we make musical harmony the next frequency.
Is it? So 3 to 4, 4 to 5, five to about 5.9 to almost six.
5.92 to 6.9, I think almost seven.
So the first six frequencies that you hear of the of a circular plate vibrating are almost in whole number ratios to each other.
And this is why if you have a timpani, which has a circular shape to it, it has a note to it.
You can hear a note because you're actually hearing the harmonics that you are hearing when a violin string is vibrating.
While intriguingly the note that you hear is not one of the notes that it's
vibrating at because your ear is tricked into putting the bottom fundamental.
So we've heard two to 3 to 4 to 5 to 6, but actually your ear fills in and you hear the bottom one, which isn't vibrating at all.
So certainly these finding plays will be important for understanding the creation of musical instruments.
But the same sort of idea what is at halt here is discovering something called the organ values.
These are the frequencies of an operator controlling these vibrations.
And the idea of an eigenvalue. Some of you students here from school may be studying matrices.
And matrices have these things called eigenvalues attached to them.
And they're really at the heart of understanding many things across the physical world.
So, for example, in quantum physics, the idea of these particular vibrations of a particular operator, just as in here,
we went through a set number of variation frequencies which produce these patterns in quantum physics as well.
The particular frequencies which you get these energy levels at is how we understand how an atom is working.
In particular, if, you know, computers are getting smaller and smaller until we're hitting the quantum world.
And what very often what we're having to understand is a kind of small area where an electron is trapped.
And we have to understand the resonances of that electron. And, again,
those particular waveforms that will control the quantum behaviour of that
electron in this little stadium will be very similar to the current NI plates.
And it's understanding the different frequencies, which will give us an understanding of how that electron is behaving.
But it's not just very small things like quantum physics. I discovered that the understanding of earthquakes in Mexico, a particular lake,
if you look at this particular shape of that lake and understand the flattening patterns that will appear in something of that shape,
helps you to understand the fault lines that are lying beneath that lake.
And in an area very close to my own, I study symmetry, but I also very interested in number theory in prime numbers.
One of the ways we believe we can understand prime numbers is using something called the Riemann Zeta function.
And we believe the way the places where this function outputs zero are going to give us the understanding of how the primes are laid out.
It's a kind of landscape and the point it points at sea level in this landscape of the DNA, which will explain the primes.
We believe that these are also eigenvalues of something, their resonances there,
like the moments in this flattening play where suddenly the shapes change from one to the other.
If we can understand what that that kind of thing is, whose frequencies are controlling the these points at sea level?
It might tell us about one of our biggest mysteries, namely understanding prime numbers.
But we're using these particular plates, the idea of putting several of these together,
because we were intrigued to use this as a way of perhaps generating some new patterns.
Traditionally, crab NI plates are shown one at a time, and you see these wonderful symmetries appearing.
But we were wondered what would happen if you put quantity plates together with different frequencies operating on different plates.
Sometimes the pattern seemed to be asking to be propagated across the plates.
And so our project this is our first experiment, your ask kind of soft launch and kind of experimenting with this idea.
We're going to you might have seen as you, some of you when you came in,
we spent the afternoon with 16 of these plates connected together in a four by four grid.
And we've been altering the frequencies and just seeing whether we can generate some new patterns.
And this is a collaboration with an artist, Richard RIESS, who's here in the audience who runs something called A Pattern Foundry.
And he's been curating patterns by different artists and patterns of his own.
And we've created a pattern together, this second pattern along something we call the ghost tile.
It's a little variation on let's see whether I guess I'm God, it's a variation on one of the Alhambra tiles.
And the idea is actually the the three sides of this tile are musical notes, which are the octave and the perfect faith.
But now we're looking at these plates to see whether we can create interesting patterns.
So here's one we just started with, which is the same frequency on each plate,
but already is creating an interesting kind of tessellation across the plain.
But our challenge is to kind of use this as a kind of tool to create a sort of musical version of the Alhambra,
something which absolutely will be living in bearing almost like a musical instrument.
So thank you for those who came this afternoon little earlier to try and help us to explore these plates.
We're going to be taking this to the Cheltenham Science Festival. So having any of you around in Cheltenham.
We're going to be there for the whole weekend just playing with our plates and seeing what symmetries we can make out of sound.
Thank you.