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