>>Marcus Du Sautoy: [applause]
Well, my first love is doing mathematics and the mathematics that I really love doing is
trying to understand the world of symmetry. So, if I was cast out and I had to spend the
rest of my life in just one building somewhere around the world then it would the Alhambra
that I would choose to be banished to. Because this, for me, is a real palace celebrating
the subject of symmetry. What I would like to do is to give you a little bit of a mathematician's
perspective on how I see this world of symmetry in a particular. What I see when I go around
the Alhambra. I've never thought of mathematics being a
language before, but actually it's an incredibly powerful language. It's a language which helps
you to understand where we've come from, what's going to happen next. Symmetry is in a sense
a language which nature uses to communicate. If you take a bee in the garden, a bee has
very bad vision. It can't judge distances, it sees the world in black and white but what
it can pick out is shapes with symmetry because it knows that that's likely to be a flower
with sustenance. The flower itself also needs to have the bee coming to propagate its genetic
heritage. So, in some ways, the symmetry is being used like a billboard, communicating
between the bee and the flower. In fact, it's said that research has found that the more
symmetrical a flower, the sweeter the nectar inside the flower. Humans too; we also use
symmetry as a way to communicate information. Here are two faces which I've made artificially
symmetrical down below. Now, if I ask you which of these do you find more beautiful
most people are drawn to the artificially symmetrical ones. Why is that? Because actually
it's communicating information. It's very difficult to make symmetry so if you can make
your face perfectly symmetrical; you're actually giving a signal. You are telling somebody
that your DNA, your genetics is very good or your upbringing is very good and you're
going to make a good mate. Plato got something which is really at the
heart of modern science, which is that you find symmetry, the mathematics of symmetry,
underpinning so many different bits of the scientific world. So, for example, molecules,
well, crystals are possible. If you understand symmetry, you understand why crystals which
may be made out of different things behave in a similar way. In biology actually, you
don't want to get a symmetrical object because they're rather nasty. A virus is actually
symmetrical and it uses its symmetry as a path or way to propagate itself very quickly
because it's a very efficient program to make more copies of itself. So, for example, the
H1N1 virus was in fact a symmetrical object. So, understanding its symmetry gives one some
understanding of what it might do. But it's not just scientists who are interested
in symmetry. The artists too have a fascination with this world of symmetry, but they have
a slightly more ambiguous attitude towards it. Here's Thomas Mann writing in "The Magic
Mountain" about symmetry. He has a character who describes a snowflake and says, "I shuddered
at its perfect precision, found it deathly, the very marrow of death." I think for many
artists symmetry is rather constricting because it sort of forces you to do certain things
so I think there's a little bit of tension with this world of symmetry. Of all the buildings
in the world, I think the one which really celebrates symmetry is the Alhambra but of
course it's on the walls really, I think, where the beautiful symmetrical games are
being played. And for me as a mathematician this palace really poses all the interesting
questions about symmetry. What exactly is symmetry? If I showed my son this wall here
and asked him does it have any symmetry and he said no, it doesn't look symmetrical at
all but what he was looking for was left right reflectional symmetry. This doesn't have any
left right reflectional symmetry but it does have symmetry. But what does it mean to have
symmetry? Also, how can I say, articulate, that two of these walls have the same symmetries?
How can I say if the Moorish artists discovered all the symmetries that are possible on the
walls in the Alhambra? Now, actually, to be able to articulate these ideas, the Moors
didn't have this sophisticated language to be able to talk about these questions. In
fact, it took a mathematician called Évariste Galois, who is a French revolutionary who
discovered a language to be able to articulate the games that are going on in the Alhambra.
Now, Évariste Galois had an incredibly dramatic life. He was in fact shot in a duel at the
age of 20 over love and politics it said. But already by the age of 20, this young mathematician
had already cracked one of the great mathematical problems at the time which gave him this insight
into how to understand symmetry. That language he left for us, a language that we call group
theory, has enabled us to actually navigate this world of symmetry. This 18 year old boy
had developed a new way of looking at symmetry which enables us to answer those questions
about the Alhambra. And what Galois realizes is that really symmetry is like the magic
trick moves. What can I do to these tiles? I can move them until suddenly they all line
up again and I can put them down exactly where they were before. So, I like to describe symmetry
as the magic trick move. If you close your eyes and I pick up these tiles and I move
them around and put them back down again in a different way, you open your eyes and they
don't look like they've moved so that is what symmetry is for Galois. Actually, it's interesting,
because for mathematicians, symmetry is actually something with a lot of motion in it. It's
got life; it's got movement, unlike Thomas Mann who thought it was very deathly. For
mathematicians, it's the moves that you can make to the wall which are the key to symmetry.
But what Galois realizes is not just the individual moves, but how they interact with each other
which is key. So, I'm going to push you a little bit mathematically now so brace yourselves,
but I want to explain this language that Galois developed to explore the symmetries of these
two objects. So, two simple objects. I've got a six pointed star with a little twist
on it and a triangle. So the magic trick moves, the symmetries, are the things that I can
do to these objects which leaves them exactly where they were so first of all, I could rotate
it by a sixth of a turn and it looks like it did before. Okay, I can rotate it by a
third of a turn and it still looks like it did before I moved it, or I can rotate 180
degrees, half a turn, so that's three symmetries or I could move all the way around to two
thirds of a turn, a fifth symmetry takes me five sixth of a turn around so there are five
different symmetries that I can do to this starfish. Now actually there's a sixth symmetry
that Galois identified here. This sixth symmetry is just leaving something where it is so that…
it's strange. It means everything has one symmetry which is just picking it up and putting
it down again. Now, it seems a bit silly but actually this is as powerful as the concept
of the number zero and we heard earlier about the Arabs, actually the Indians were the first
to develop the idea of zero. Beforehand, the Europeans said, "Why do we need a number for
zero? If you're not counting anything, you don't need a number." But the Indians realized
the power of zero to actually make advances in mathematics. So, here again this is like
the zero symmetry, if you like to think about it. So, there are six symmetries that this
starfish has. Let's take the triangle. I can rotate that by a third of a turn clockwise
or a third of a turn anti-clockwise and I've got, now I can reflect this as you said. I
can reflect in the line through 'x' or I can reflect in the line through 'y' or I can reflect
in the line through 'z.' So, five symmetries again, plus the extra symmetry where I'm just
picking it up and put it back down again. So, both of these objects have six symmetries.
So, should we say they have the same symmetries because they have the same amount of symmetry?
Well, now you feel there's some difference between these objects, but how can we articulate
why the symmetries of these things really are different?
For Galois, what he realized is, it's the interactions of these symmetries which distinguish
them. If I do one magic trick move followed by another one, the combination is a third
symmetry so you can actually develop a kind of algebra. If I do one thing followed by
another, what's the answer? It's a third symmetry. So, here I'm going to do a little bit of changing.
This is the power of the language of mathematics. Galois changed geometry into a kind of algebra
which tells me how the symmetries interact. So, what I'm going to do, I've given names
to each of the symmetrical moves so 'a' is just leaving the thing where it is. Capital
'B' is the turn where the little yellow dot rolls down to the lowercase 'b' so 'c' is
a third of a turn, 'd' is half a turn, 'e' around to here, and 'f' there. Now, what happens
if I do a sixth of a turn followed by a third of a turn? So I'm going to do rotation 'b'
followed by rotation 'c', which is a third of a turn, and the combined effect of those
is if I'd just done half a turn in one go. So, this table shows me how the symmetries
interact with each other and this was the inspirational move of Galois is that this
is key to understanding the symmetries of this object. If I do this in a different order,
doesn't matter really, do the third of a turn followed by a sixth of a turn, it still ends
up in the same place. So, there's some symmetry in the way the symmetries interact, but that's
not true for the triangle. Let's do two symmetries of the triangle. We're
going to rotate by a third of a turn anti-clockwise then reflect in 'x'. The combined effect is
if I've just reflected in 'z' but now do these in a different order; do the reflection first
and then rotate by a third of a turn and it ends up in a different place. It's as if I'd
reflected in the line through 'y.' So, the way these symmetries interact enables me to
distinguish the symmetries of the triangle from the symmetries of that six pointed starfish.
And in fact, Galois developed a sort of set of rules for how the symmetries interact.
It's a little bit like the rules for Sudoku. I don't know whether any of you do Sudoku's
here but remember you are not allowed the same number in any column or any row. Well,
that's the same for symmetries as well. You are not allowed any symmetry in the same row
here, in the main body of the table, or in the same column. And Galois developed some
rules for the way symmetries must interact which enabled him to prove actually that if
an object has six symmetries, then either its symmetries behave in the same way of the
triangle or the six pointed starfish. So, it enabled him to abstract the idea of symmetry
underlying something and this is perfect now when we go to the Alhambra. Because if we
look at the tiles on the walls in the Alhambra, how can we say when two of those walls have
the same underlying symmetry? Now, they might look geometrically very different but Galois's
language enables to say actually the symmetries of these two walls are the same symmetries,
although they look very different. So, we've got this beautiful walls with these little
twisted triangles and here I've got this six pointed star with this kind of 'z'. So let's
look at the symmetries of these two walls. Remember, symmetry are the magic trick moves
so I can pick up these third tiles, rotate them by a sixth of a turn around the point
where all of these little triangles meet, and then they all line up again so that's
one symmetry, a sixth of a turn around there, but I've got another symmetry. I can rotate,
I can fix the point in the middle of one of these tiles and rotate by a third of a turn
there and they all line up perfectly. Now, there's another rather hard symmetry to find
but if you actually fix halfway along an edge, you can actually fix that point, rotate all
of them by 180 degrees, and then suddenly all of them fit perfectly down on top of each
other. It's quite fun. I mean, this is the fun thing about going to the Alhambra. At
least, for me as a mathematician. [laughs]
It's trying to find, well, where are all of these points where you fix things, rotate
them, or reflect them, and if you go now. So, you've got those points there but you
see the same symmetries happening on this wall which looks very different. Rotate by
a sixth of a turn around the point in the middle of the six pointed star and everything
matches up. The third of a turn is actually where all of these 'z' pieces meet so again
rotate by a third of a turn and eventually, ping, third of a turn and everything matches
up. The half one is really difficult to find. It's halfway between the two six pointed stars.
Here if you rotate everything, you have to rotate it by 180 degrees before it all matches
up. So, although these walls look very different, Galois's language says but no, the symmetries
will interact in exactly the same way. In fact, you could put one wall on top of the
other and whatever symmetry you did to one wall would give you a symmetry of the wall
underneath. So, this actually, this group of symmetries, we give the name 632 because
of these three different sorts of rotations. Here are two different, three different, well,
this is actually the tiles that we met when we entered the Alhambra. This is actually
the ceiling in that first room in the Alhambra and this is the tiles on the floor. Now again,
these pictures look very different but the underlying symmetries, thanks to Galois's
language, we can say are actually the same. And this group of symmetries we call 4,4,2.
Nothing to do with football but because of the fact that there's a place where I can
rotate by a quarter of a turn, a different place here, and here again somewhere we have
to rotate by half a turn. So, Galois's language is absolutely extraordinary. I think it does
for symmetry what the idea of number did for us many years ago. I mean, the concept of
number is very abstract. For example, I've got one, two, three, four, five, six people
on the front here and they're sitting on six chairs. Now, the people are very different
from the chairs, you'll be glad to hear, but the abstract idea of the number six is common
to both of them. But we've abstracted this idea of number which is common to things which
have very different physical manifestations. Galois did the same thing for symmetry. These
walls look very different but he's found a way to abstractly say, "No, the symmetries
underlying them are actually the same." And using those kind of Sudoku rules, it wasn't
Galois but mathematicians at the end of the 19th century were able to then say well, how
many different symmetrical games are there? If these are all the same, how many different
ones are there? But using this language of Galois, we discovered there are actually only
17 different symmetrical designs that you can make in the Alhambra and actually one
of the games we play in that chapter of the month of October was to go and see did the
Moorish artists discover examples of all 17. They are incredibly inventive with the way
that they tried different shapes and things like that. Very often they repeated things
like the 6,3,2 or 4,4,2. In fact, it took me several days but we managed to find examples
of all 17 there but I think it's the power of mathematics to say that however hard you
try, you are not going to be able to make a new symmetrical wall which is going to have
symmetries different from these 17. So, I can describe this symmetrical tile using numbers.
It's a tile which has points at 0,0 and then you move one step north to 0,1 and one step
east to 1,0 and extreme point at 1,1. Three dimensions that I can use the height above
sea level so I use three coordinates, 0,0,0. I can describe all of the points on a three
dimensional cube. Now, if I want to make symmetrical shapes to tile my Alhambra in higher dimensions,
I run out of pictures. We don't have pictures of four dimensional shapes, but the wonderful
thing about Descartes Dictionary, this power of the language of mathematics, is that actually
I can describe for you a four dimensional cube. It starts at point at 0,0,0,0 and then
it has edges running to four points at 1,0,0,0, 0,1,0,0, 0,0,1,0, 0,0,0,1, all the way through
to the extreme point at 1,1,1,1. I think that is the power of mathematics, you know? Although
I didn't realize my dream to become a diplomat. Come here to Brussels to represent my country
or whatever. Actually, I ended up learning another language which allowed me access to
a very strange country. In a way, I say I'm an ambassador for the world of mathematics,
so that's what I spend my research doing. It's trying to explore this world of symmetry.
What other, we know there are 17 in the Alhambra but what about the Alhambra in four dimensions,
five dimensions? So, actually that's what I spend my life doing, creating symmetrical
objects. The exciting things for me about mathematics is although we understand the
symmetries in the Alhambra, there are still many things about symmetry we don't understand
and I think that's what for me makes mathematics so exciting. It's a living subject because
of all the things we don't understand. I actually always love coming back to this quote that
my professor in Japan that I work with told me about because I think it sums up why mathematics
is a living subject. "In everything… uniformity is undesirable. Leaving something incomplete
makes it interesting, and gives one the feeling that there is room for growth…" In some
ways, it's all the unsolved problems, the symmetries I don't understand, which make
mathematics such an exciting, living subject. So, thank you very much.
[applause]