Symmetry realitys riddle Marcus du Sautoy
on the 30th of May 1832 a gunshot was
heard ringing out across the 13th
arrondissement in Paris peasant who was
walking to market that morning ran
towards where the gunshot had come from
and found a young man writhing in agony
on the floor clearly shot by a dueling
wound the young man’s name was ever East
galois he was a well-known revolutionary
in Paris at the time
Galois was taken to the local hospital
where he died the next day in the arms
of his brother and the last words he
said to his brother where don’t cry for
me Alfred I need all the courage I can
muster to die at the age of 20 that
wasn’t in fact revolutionary politics
for which Galois was famous but a few
years earlier while still at school he’d
actually cracked one of the big
mathematical problems at the time and he
wrote to the academicians in Paris
trying to explain his theory but the
academicians couldn’t understand
anything that he wrote this is how he
wrote most of his mathematics so the
night before that Joule he realized that
this possibly is his last chance to try
and explain his great breakthrough so he
stayed up the whole night writing away
trying to explain his ideas and as the
dawn came up and he went to meet his
destiny he left this pile of papers on
the table for the next generation maybe
the fact these stayed up all night doing
mathematics was the fact that he was
such a bad shot that morning I got
killed and but contained inside those
documents was a new language a language
to understand whether the most
fundamental concepts of science namely
symmetry now symmetry is almost nature’s
language it helps us to understand so
many different bits of the scientific
world for example molecular structure
what crystals are possible we can
understand through the mathematics of
symmetry in microbiology you really
don’t want to get a symmetrical object
because they’re generally rather nasty
the swine flu virus at the moment is a
symmetrical object and it uses the
efficiency of symmetry to be able to
propagate itself so well
but a larger-scale of biology actually
symmetry is very important because
actually communicates genetic
information I’ve taken two pictures here
and I’ve made them artificially
symmetrical and if I ask you which of
these you find more beautiful you’ll
probably be drawn to the lower two
because it’s hard to make symmetry and
if you can make yourself symmetrical
you’re sending out a sign that you’ve
got good genes you’ve got a good
upbringing and therefore you’ll make a
good mate so symmetry is a language
which can help to communicate genetic
information symmetry can also help us to
explain what’s happening in the Large
Hadron Collider in CERN or what’s not
happening in the Large Hadron Collider
in CERN to be able to make predictions
about the fundamental particles we might
see there it seems that there are all
facets of some strange symmetrical shape
in a higher dimensional space and I
think Galileo summed up very nicely the
power of mathematics to understand the
scientific world around us
he wrote the universe cannot be read
until we have learned a 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
geometric figures without which means it
is humanly impossible to comprehend a
single word but it’s not just scientists
who interested in symmetry artists to
love to play around with symmetry they
also have a slightly more ambiguous
relationship with it here’s Thomas Mann
talking about symmetry in the magic
mountain he has a character describing
the snowflake and he says he shuddered
at its perfect precision he added
deathly the very marrow of death but
what I just like to do is to set up
expectations of symmetry and then break
them and a beautiful example of this I
found actually when I visited a
colleague of mine in Japan professor
Cora Kawa and he took me up to the
temples in Nikko and just after this
photo was taken we walked up the stairs
and the Gateway you see behind has eight
columns with beautiful symmetrical
designs on sever them them are exactly
the same and the eighth one is turned
upside down and I said to professor
Kurokawa Wow the architects must have
been really kicking themselves and they
realized that you know they made the
mistake and put this one upside down he
said no no no it was a very deliberate
act and he refer me to this lovely quote
from the Japanese essays in idleness
from the 14th century in which the SAS
in everything uniformity is undesirable
leaving something incomplete makes it
interesting and gives one the feeling
that there is room for growth even when
building the imperial palace they always
leave one place unfinished but if I had
to choose one building in the world to
be cast out on the desert island to live
the rest of my life being an addict of
symmetry I would probably choose the
Alhambra in Granada this is a palace
celebrating symmetry recently I took my
family we do this rather kind of nerdy
mathematical trips so which my family
love this is my son Tamir you can see
he’s really enjoying our mathematical
trip to the Alhambra but I wanted to try
and enrich him I think one of the
problems about school mathematics is
it’s it’s it doesn’t look at how
mathematics is embedded in the world we
live in so I wanted to open up his eyes
up to how much symmetry is running
through the Alhambra and you see it all
really immediately you go in the
reflective symmetry in the water but
it’s on the walls where all the exciting
things are happening the Moorish artists
would deny the possibility to draw
things with Souls
so they explored a more geometric art
and so what is symmetry and the Alhambra
somehow asks all of these questions what
is symmetry when a two of these walls do
they have the same symmetries can we say
whether they discovered all of the
symmetries in the Alhambra and it was
Galois who produced a language to be
able to answer some of these questions
the Galois symmetry unlike for Thomas
Mann which was something still and
deadly the Galois symmetry was all about
motion what can you do to a symmetrical
object move it in some way so it looks
the same as before you moved it I like
to describe it as the magic trick moves
what can you do to something you close
your eyes I do something put it back
down again
and it looks like it did before it
started so for example the walls in the
Alhambra I can take all of these tiles
and fix them at the yellow place rotate
them by 90 degrees put them all back
down again and they fit perfectly down
there and if you open your eyes again
you wouldn’t know that they’ve moved but
it’s the motion that really
characterizes the symmetry inside the
Alhambra but it’s also about producing a
language to describe this and the power
of mathematics is often to change one
thing into another to change geometry
into language
take you through perhaps push you a
little bit mathematically so brace
yourselves push you a little bit to
understand how this language works which
enables us to capture what is symmetry
so let’s take these two symmetrical
objects here let’s take the twisted six
pointed star fish what can I do to this
starfish which makes it look the same
well there I rotated it by a sixth of a
turn and still it looks like it did
before I started I can rotate by a third
of a turn or a half a turn and put it
back down on its image or 2/3 of a turn
and a fifth symmetry I can rotate it by
five sixth of a turn and those are
things that I can do to the symmetrical
object which make it look like it did
before I start it now for Galois there
was actually a sixth symmetry can
anybody think what else I could do to
this which would leave it like it did
before I started I can’t flip it because
I put a little twist on it term tie it’s
got no reflective symmetry but what I
could do is just leave it where it is
pick it up and put it down again and for
Galois this was like the zeroth symmetry
actually the invention of this number
zero was a very modern concept 7th
century AD by the Indians
it seems mad to talk about nothing and
this is the same idea this is a
symmetrical to everything has symmetry
where you just leave it where it is so
this object has six symmetries and what
about the triangle well I can rotate by
third of a turn clockwise or a third of
a turn anti-clockwise but now this has
some reflectional symmetry I can reflect
it in the line through X or the line
through Y or the line through Z five
symmetries and then of course the zero
symmetry where I just pick it up and
leave it where it is so both of these
objects have six symmetries now I’m a
great believer that mathematics is not a
spectator sport and you have to do some
mathematics in order to really
understand it so here’s a little
question for you and I getting a look of
a prize at the end of my talk for the
person who gets closest to the answer
the Rubik’s Cube how many symmetries
does a Rubik’s Cube have how many things
can I do to this object and put it down
so it still looks like a cube okay so I
want you to think about that problem as
we go on and count how many symmetries
there are and there’ll be a prize to the
person who gets closest at the end
but let’s go back down to symmetries
that I got for these two objects what
Galois I realize it isn’t just the
individual symmetries but how they
interact with each other which really
characterizes the symmetry of an object
if I do one magic trick move followed by
another the combination is a third magic
trick move and here we see Galois
starting to develop a language to see
the substance of the things unseen the
sort of abstract idea of the symmetry
underlying this physical object for
example what do I turn the starfish by a
sixth of a turn and then a third of a
turn so I given names the capital
letters ABCDE F are the names for the
rotations so be for example rotates the
little yellow dot to the be on the
starfish and so on so what if I do B
which is a sixth of a turn followed by C
which is a third of a turn well let’s do
that a sixth of a turn followed by a
third of a turn the combined effect s is
if I just rotated it by half a turn in
one go so the little table here records
how the algebra of these symmetries work
I do one followed by another the answer
is its rotation D half a turn what if I
did it in the other order would it make
any difference well let’s see let’s do
the third of the turn first and then the
sixth of a turn of course it doesn’t
make any difference it still ends up at
half a turn and there’s some symmetry
here in the way the symmetries interact
with each other but this is completely
different to the symmetries of the
triangle let’s see what happens if we
two two symmetries with a triangle one
after the other do a rotation by a third
of a turn anti-clockwise and reflect in
the line through X well the combined
effect is if I just done the reflection
in the line through Z to start with now
let’s do it in a different order let’s
do the reflection in X first followed by
the rotation by a third of a turn
anti-clockwise the combined effect the
triangle ends up somewhere completely
different it’s as if it wasn’t reflected
in the line through Y now it matters
what order you do the operations in and
this aware nabel’s us to distinguish why
the symmetries of these objects they
both have six symmetries so why
shouldn’t we say they have the same
symmetries but the way the symmetries
interact enable us we’ve now got a
language distinguish why these
trees are fundamentally different and
you could try this when you go down the
pub later on take a beer mat and rotate
it by third quarter of a turn then flip
it and then do it in the other order and
the picture will be facing in the
opposite direction now Galois produced
some laws for how these tables how
symmetries interact it’s always like
little Sudoku tables you don’t see any
symmetry twice any row or column and
using those rules he was able to say
that there are in fact only two objects
with six symmetries and they’ll be the
same as the symmetries of the triangle
or the symmetries of the six pointed
star fish I think this is an amazing
development it’s almost like the concept
of number being developed for symmetry
in the frontier I’ve got one two three
people sitting on one two three chairs
the people and the chairs are very
different but the number the abstract
idea of the number is the same and we
can see this now we go back to the walls
in the Alhambra here are two very
different walls very different geometric
pictures but using the language of
Galois we can understand that the
underlying abstract symmetries of these
things are actually the same for example
let’s take this beautiful wall with the
triangles did a little twist on them you
can rotate them by a sixth of a turn if
you ignore the colors we’re not matching
up the colors but the shapes match up if
I rotate by sixth of a turn around the
point where all the triangles meet what
about the center of a triangle I can
rotate my third of a turn around the
center of the triangle and everything
matches up then there’s an interesting
place halfway along an age where I can
rotate by 180 degrees and all the tiles
match up again so rotate along half way
along the edge and they all match up now
let’s move to the very different-looking
wall in the Alhambra and we find the
same symmetries here and the same
interaction so there was a sixth of the
turn a third of a turn with as nth
pieces meet and then the half a turn is
halfway between the six pointed stars
and although these walls look very
different Galois has produced a language
to say that in fact the symmetry is
underlying these are exactly the same
and it’s a symmetry we call six three
two here’s another example in the
Alhambra this is a wall a ceiling and a
floor they all look very different but
this language allows us to say they are
representations of
same symmetrical abstract object which
we call 4-4-2 nothing to do with
football but because of the fact that
there are two places where you can
rotate by a quarter of a turn and one by
half a turn now this part of the
language is even more because Galois can
say did the Moorish artists discover all
of the possible symmetries on the walls
in the Alhambra and it turns out they
almost did you can prove using Galois
language there are actually only 17
different symmetries that you can do in
the walls in the Alhambra and they if
you try and produce a different wall
with its 18th one it will have to have
the same symmetries as one of these 17
but these are things that we can see and
the power of Galois mathematical
language is it also allows us to create
symmetrical objects in the unseen world
beyond the two-dimensional
three-dimensional all the way through to
the 4 5 or infinite dimensional space
and that’s where I work I create
mathematical objects symmetrical objects
using Galois z– language in very high
dimensional spaces so I think it’s a
great example of things unseen which the
power of mathematical language allows
you to create so like Galois I stayed up
all last night creating a new
mathematical symmetrical object for you
and I’ve got a picture of it here well
unfortunate isn’t really a picture if I
could have my board at the side here
great excellent
here we are this is unfortunately I
can’t show you a picture of this
symmetrical object but here is the
language which describes how the
symmetries interact now this new
symmetrical object does not have a name
yet now people like getting any names on
things on sort of craters on the moon or
new species of animals so I’m going to
give you the chance to get your name on
a new symmetrical object which hasn’t
been named before and this thing species
died away and moons kind of get hit by
meteors and explode but this
mathematical object will live forever it
will make you immortal in order to win
your win this symmetrical object what
you have to do is to answer the question
I asked you at the beginning
how many symmetries of the Rubik’s Cube
have ok I’m going to sort you out
rather than you all shouting out I want
you to count how many digits there are
in that number okay if you’ve got it as
a factorial you have to expand the
factorial
okay now if you want to play I want you
to stand up okay if you think you can
you’ve got an estimate for how many
digits right we’ve already got one
competitor here yeah you all stay down
he wins it automatically okay excellent
so we’ve got four here five six great
excellent after that I should get us
going
all right anybody with five or less
digits you’ve got to sit down because
you’ve underestimated five or less
digits so a hundred thousands of
thousands you’ve got to sit down 60
digits or more you’ve got to sit down
you’ve overestimated 20 digits or less
sit down Oh 20 how many digits are there
in your number two so you sort of sat
down earlier let’s have the other ones
who said oh they said the other ones who
sat sat down during the 20 up again okay
if I told you 20 or less stand up
because we’re this one I think there are
a few here you’ve just said the people
who just last sat down okay how many
digits do you have in your number ah ha
ha how many 21 ok good how many do have
a new one 18 so it goes to this lady
here 21 is the closest they actually has
the number of symmetries in the Rubik’s
Cube has 25 digits so now I need to name
this object so what is your name I need
your surname groups the symmetrical
objects generally spell it for me G H e
Z now so2 s already been used as you in
the mathematical language so you can’t
have that so gets there we go that’s
your new symmetrical object you are now
immortal
and if you’d like your own symmetrical
object I have a project so raising money
for a charity in Guatemala where I will
stay up all night and devise an object
for you for a donation to this charity
to help kids get into education in
Guatemala and I think what drives me is
a mathematician are those things which
are not seen the things that we haven’t
discovered and it’s all the unanswered
questions which make mathematics a
living subject and I always come back to
this quote from the Japanese essays
denying idleness in everything
uniformity is undesirable leaving
something incomplete makes it
interesting and gives one the feeling
that there is room for growth thank you