Marcus du Sautoy: Symmetry, reality's riddle

Prevoditelj: Sanda L Recezent: Ivan Stamenković

On the 30th of May, 1832,

30. svibnja 1832.

a gunshot was heard

čuo se pucanj

ringing out across the 13th arrondissement in Paris.

kako odzvanja u 13. pariškom okrugu.

(Gunshot)

(Pucanj)

A peasant, who was walking to market that morning,

Tog je jutra jedan seljak, na svom putu prema tržnici,

ran towards where the gunshot had come from,

potrčao prema mjestu pucnja

and found a young man writhing in agony on the floor,

i zatekao na podu mladića u agoniji,

clearly shot by a dueling wound.

očigledno ustrijeljenog u dvoboju.

The young man's name was Evariste Galois.

Ime tog mladića bilo je Evariste Galois.

He was a well-known revolutionary in Paris at the time.

Bio je poznati pariški revolucionar tog vremena.

Galois was taken to the local hospital

Galois je prebačen u mjesnu bolnicu,

where he died the next day in the arms of his brother.

gdje umire sljedećeg dana u naručju svoga brata.

And the last words he said to his brother were,

Posljednje riječi koje je rekao svom bratu bile su,

"Don't cry for me, Alfred.

„Ne plači zbog mene, Alfrede.

I need all the courage I can muster

Potrebna mi je sva moguća snaga

to die at the age of 20."

kako bih umro u dvadesetoj."

It wasn't, in fact, revolutionary politics

Zapravo, Galois se nije proslavio

for which Galois was famous.

zbog neke revolucionarne politike.

But a few years earlier, while still at school,

Nekoliko godina ranije, dok je još bio u školi,

he'd actually cracked one of the big mathematical

riješio je jedan od velikih

problems at the time.

matematičkih problema tog vremena.

And he wrote to the academicians in Paris,

Pisao je pariškim akademicima,

trying to explain his theory.

pokušavajući objasniti svoju teoriju.

But the academicians couldn't understand anything that he wrote.

Ali akademici nisu razumjeli ništa od onoga što je napisao.

(Laughter)

This is how he wrote most of his mathematics.

(Smijeh)

Ovako je pisao većinu svoje matematike.

So, the night before that duel, he realized

Noć prije dvoboja, shvatio je

this possibly is his last chance

da mu je to možda zadnja prilika

to try and explain his great breakthrough.

kako bi razjasnio svoje veliko otkriće.

So he stayed up the whole night, writing away,

Ostao je budan cijelu noć, pišući

trying to explain his ideas.

i pokušavajući objasniti svoje ideje.

And as the dawn came up and he went to meet his destiny,

Dolaskom zore otišao je ususret svojoj sudbini,

he left this pile of papers on the table for the next generation.

ostavivši hrpu papira na svom stolu za buduću generaciju.

Maybe the fact that he stayed up all night doing mathematics

Možda je to što je probdio noć rješavajući matematiku,

was the fact that he was such a bad shot that morning and got killed.

doprinijelo tome da to jutro tako loše puca i pogine.

But contained inside those documents

Ali, sadržaj u tim papirima

was a new language, a language to understand

bio je nov jezik, jezik za razumijevanje

one of the most fundamental concepts

jednog od osnovnih koncepata

of science -- namely symmetry.

znanosti -- a to je simetrija.

Now, symmetry is almost nature's language.

Simetrija je, gotovo, jezik prirode.

It helps us to understand so many

Pomaže nam shvatiti

different bits of the scientific world.

toliko različitih komadića znanstvenog svijeta.

For example, molecular structure.

Primjerice, molekularnu strukturu.

What crystals are possible,

Kakvi kristali su mogući,

we can understand through the mathematics of symmetry.

možemo shvatiti kroz matematiku simetrije.

In microbiology you really don't want to get a symmetrical object,

U mikrobiologiji zaista ne želite dobiti simetričan objekt,

because they are generally rather nasty.

jer su načelno dosta gadni.

The swine flu virus, at the moment, is a symmetrical object.

Virus svinjske gripe, trenutno, je simetričan.

And it uses the efficiency of symmetry

A koristi se učinkovitošću simetrije

to be able to propagate itself so well.

kako bi se tako dobro širio.

But on a larger scale of biology, actually symmetry is very important,

I u širim razmjerima biologije simetrija je jako važna,

because it actually communicates genetic information.

jer, zapravo, prenosi genetske podatke.

I've taken two pictures here and I've made them artificially symmetrical.

Snimio sam dvije slike a onda ih umjetno napravio simetričnima.

And if I ask you which of these you find more beautiful,

Pitam li vas koje su vam ljepše,

you're probably drawn to the lower two.

vjerojatno će vas privući donje dvije.

Because it is hard to make symmetry.

To je zato jer je teško postići simetriju.

And if you can make yourself symmetrical, you're sending out a sign

Učinite li sebe simetričnima, šaljete signal

that you've got good genes, you've got a good upbringing

da su vaši geni dobri, da ste dobro odgojeni,

and therefore you'll make a good mate.

što navodi da ćete biti kvalitetan partner.

So symmetry is a language which can help to communicate

Simetrija je, stoga, jezik koji vam pomaže

poslati genetsku poruku.

genetic information.

Symmetry can also help us to explain

Simetrija nam isto pomaže objasniti

what's happening in the Large Hadron Collider in CERN.

što se događa u Velikom hadronskom sudaraču u CERN-u.

Or what's not happening in the Large Hadron Collider in CERN.

Ili što se ne događa u Velikom hadronskom sudaraču u CERN-u.

To be able to make predictions about the fundamental particles

Kako bismo predvidjeli koje elementarne čestice

we might see there,

bismo mogli tamo vidjeti,

it seems that they are all facets of some strange symmetrical shape

čini se da su to sve aspekti nekog čudnog simetričnog oblika

in a higher dimensional space.

u prostoru širih dimenzija.

And I think Galileo summed up, very nicely,

Mislim da je Galileo jako dobro izložio

the power of mathematics

moć matematike

to understand the scientific world around us.

u razumijevanju znanstvenog svijeta oko nas.

He wrote, "The universe cannot be read

Napisao je: „Svemir se ne može pročitati

until we have learnt the language

dok nismo naučili jezik

and become familiar with the characters in which it is written.

i upoznali se s pismom kojim je napisan.

It is written in mathematical language,

Napisan je matematičkim jezikom,

and the letters are triangles, circles and other geometric figures,

a slova su mu trokuti, krugovi i drugi geometrijski oblici,

without which means it is humanly impossible

bez kojih je nemoguće

to comprehend a single word."

shvatiti ijednu riječ."

But it's not just scientists who are interested in symmetry.

Ali nisu samo znanstvenici zainteresirani za simetriju.

Artists too love to play around with symmetry.

Umjetnici se isto vole igrati simetrijom.

They also have a slightly more ambiguous relationship with it.

Također imaju i dvosmisleniji odnos s njom.

Here is Thomas Mann talking about symmetry in "The Magic Mountain."

Ovdje Thomas Mann govori o simetriji u "Čudesnoj gori".

He has a character describing the snowflake,

Njegov lik objašnjava snježnu pahulju

and he says he "shuddered at its perfect precision,

i kaže kako je "zadrhtao nad njenom savršenom preciznošću,

found it deathly, the very marrow of death."

smatra je smrtonosnom, poput same srži smrti."

But what artists like to do is to set up expectations

Umjetnici vole postaviti očekivanja

of symmetry and then break them.

simetrije i onda ih uništiti.

And a beautiful example of this

Prekrasan primjer za to

I found, actually, when I visited a colleague of mine

našao sam kad sam posjetio svog kolegu

in Japan, Professor Kurokawa.

u Japanu, profesora Kurokawu.

And he took me up to the temples in Nikko.

Odveo me je u hramove grada Nikko.

And just after this photo was taken we walked up the stairs.

Odmah nakon ove slike, uspinjali smo se stepenicama.

And the gateway you see behind

I prolaz koji vidite iza nas

has eight columns, with beautiful symmetrical designs on them.

ima osam stupova ukrašenih prekrasnim simetričnim oblicima.

Seven of them are exactly the same,

Sedam ih je potpuno jednakih,

and the eighth one is turned upside down.

a osmi je preokrenut.

And I said to Professor Kurokawa,

Rekao sam profesoru Kurokawi,

"Wow, the architects must have really been kicking themselves

sigurno su se arhitekti lupali po glavi

when they realized that they'd made a mistake and put this one upside down."

kada su shvatili da su pogriješili i ovoga preokrenuli.

And he said, "No, no, no. It was a very deliberate act."

A on kaže: "Ne, ne, ne. To je hotimično napravljeno."

And he referred me to this lovely quote from the Japanese

I onda me uputio na prekrasni citat iz japanskih

"Essays in Idleness" from the 14th century,

"Eseja o ljenčarenju" iz 14. stoljeća,

in which the essayist wrote, "In everything,

u kojima pisac kaže, "U svemu je

uniformity is undesirable.

jednoličnost nepoželjna.

Leaving something incomplete makes it interesting,

Ostavljati nešto nepotpunim, čini ga zanimljivim

and gives one the feeling that there is room for growth."

i ostavlja dojam da ima prostora za rast."

Even when building the Imperial Palace,

Čak i pri gradnji carske palače,

they always leave one place unfinished.

uvijek su ostavili jedan dio nedovršen.

But if I had to choose one building in the world

Ali kad bih izabrao jednu zgradu na svijetu

to be cast out on a desert island, to live the rest of my life,

koju bih preselio na pusti otok gdje bih proživio ostatak života,

being an addict of symmetry, I would probably choose the Alhambra in Granada.

kako sam ovisnik o simetriji, vjerojatno bih izabrao Alhambru u Granadi.

This is a palace celebrating symmetry.

Ova palača slavi simetriju.

Recently I took my family --

Nedavno sam odveo svoju obitelj --

we do these rather kind of nerdy mathematical trips, which my family love.

odlazimo na ovakve štreberske matematičke izlete koje moja obitelj obožava.

This is my son Tamer. You can see

Evo mog sina Tamera. Vidite

he's really enjoying our mathematical trip to the Alhambra.

da se stvarno zabavlja na našem matematičkom putovanju u Alhambru.

But I wanted to try and enrich him.

Želio sam obogatiti njegovo iskustvo.

I think one of the problems about school mathematics

Smatram da je problem sa školskom matematikom

is it doesn't look at how mathematics is embedded

u tome što ne promatra koliko je matematika

in the world we live in.

utkana u svijet u kojem živimo.

So, I wanted to open his eyes up to

Zato sam želio da uoči

how much symmetry is running through the Alhambra.

koliko se simetrije nalazi u Alhambri.

You see it already. Immediately you go in,

Vidi se odmah. Čim uđete,

the reflective symmetry in the water.

vidite reflektivnu simetriju vode.

But it's on the walls where all the exciting things are happening.

No, na zidovima se nalazi ono najuzbudljivije.

The Moorish artists were denied the possibility

Maurskim umjetnicima nije bilo dozvoljeno

to draw things with souls.

crtati ono što ima dušu.

So they explored a more geometric art.

Stoga su istraživali geometrijsku umjetnost.

And so what is symmetry?

Dakle, što je simetrija?

The Alhambra somehow asks all of these questions.

Alhambra, zapravo, postavlja sva ta pitanja.

What is symmetry? When [there] are two of these walls,

Što je simetrija? Ako imamo dva ovakva zida,

do they have the same symmetries?

jesu li oni jednako simetrični?

Can we say whether they discovered

Možemo li reći da su otkrili

all of the symmetries in the Alhambra?

svu simetriju u Alhambri?

And it was Galois who produced a language

Galois je taj koji je stvorio jezik

to be able to answer some of these questions.

kako bi odgovorio na neka od tih pitanja.

For Galois, symmetry -- unlike for Thomas Mann,

Za Galoisa je simetrija, za razliku od Thomasa Manna,

which was something still and deathly --

kome je ona nepomična i samrtnička --

for Galois, symmetry was all about motion.

za Galoisu je simetrija stvar gibanja.

What can you do to a symmetrical object,

Što možete učiniti simetričnom objektu,

move it in some way, so it looks the same

pomaknete li ga nekamo, da izgleda isti

as before you moved it?

kao prije pomicanja?

I like to describe it as the magic trick moves.

Volim to opisati kao pokrete mađioničarskog trika.

What can you do to something? You close your eyes.

Što možete nečemu učiniti? Zatvorite oči.

I do something, put it back down again.

Nešto učinim i spustim.

It looks like it did before it started.

Izgleda kao na početku.

So, for example, the walls in the Alhambra --

Tako, primjerice, zidovi Alhambre --

I can take all of these tiles, and fix them at the yellow place,

Mogu uzeti sve te pločice i pričvrstiti ih na žutom dijelu,

rotate them by 90 degrees,

zarotirati ih za 90 stupnjeva,

put them all back down again and they fit perfectly down there.

spustiti ih natrag i savršeno odgovaraju.

And if you open your eyes again, you wouldn't know that they'd moved.

I kad biste ponovo otvorili oči, ne biste znali da su pomaknute.

But it's the motion that really characterizes the symmetry

Ali, upravo je pomak taj koji karakterizira simetriju

inside the Alhambra.

unutar Alhambre.

But it's also about producing a language to describe this.

A radi se i o stvaranju jezika kojim bi se to opisalo.

And the power of mathematics is often

Moć matematike je često

to change one thing into another, to change geometry into language.

u promjeni jedne stvari u drugu, promjeni geometrije u jezik.

So I'm going to take you through, perhaps push you a little bit mathematically --

Stoga ću vas provesti, možda malo pogurnuti matematički

so brace yourselves --

pa se pripremite --

push you a little bit to understand how this language works,

pogurati vas malo kako biste razumjeli kako radi ovaj jezik,

which enables us to capture what is symmetry.

koji nam omogućuje usvojiti što je simetrija.

So, let's take these two symmetrical objects here.

Stoga, uzmimo ova dva simetrična predmeta.

Let's take the twisted six-pointed starfish.

Uzmimo uvijenu šesterokraku morsku zvijezdu.

What can I do to the starfish which makes it look the same?

Što mogu učiniti morskoj zvijezdi da bi izgledala isto?

Well, there I rotated it by a sixth of a turn,

Evo, rotirao sam je za šestinu okreta

and still it looks like it did before I started.

i dalje izgleda kao na početku.

I could rotate it by a third of a turn,

Mogu je zakrenuti za trećinu okreta,

or a half a turn,

ili pola okreta,

or put it back down on its image, or two thirds of a turn.

ili spustiti na njezin odraz, ili za dvije trećine okreta.

And a fifth symmetry, I can rotate it by five sixths of a turn.

I peta simetrija, mogu je zarotirati za pet šestina okreta.

And those are things that I can do to the symmetrical object

To su stvari koje mogu učiniti simetričnom predmetu,

that make it look like it did before I started.

a da izgleda kao na početku.

Now, for Galois, there was actually a sixth symmetry.

Za Galoisa je postojala i šesta simetrija.

Can anybody think what else I could do to this

Može li se itko sjetiti što bih još mogao učiniti,

which would leave it like I did before I started?

a da bude kao na početku?

I can't flip it because I've put a little twist on it, haven't I?

Ne mogu je preokrenuti jer sam stavio oznaku, zar ne?

It's got no reflective symmetry.

Nema refleksivnu simetriju.

But what I could do is just leave it where it is,

Ali bih je mogao ostaviti ovdje gdje jest,

pick it up, and put it down again.

podignuti i ponovno spustiti.

And for Galois this was like the zeroth symmetry.

Galoisu je to bilo poput nulte simetrije.

Actually, the invention of the number zero

Zapravo, izum broja nula

was a very modern concept, seventh century A.D., by the Indians.

zapravo je moderan koncept Indijaca iz sedmog stoljeća p. K.

It seems mad to talk about nothing.

Čini se ludim govoriti o ničemu.

And this is the same idea. This is a symmetrical --

A ovo je isto to. To je simetrično --

so everything has symmetry, where you just leave it where it is.

stoga sve ima simetriju, samo to ostavite gdje jest.

So, this object has six symmetries.

Dakle, ovaj predmet ima šest simetrija.

And what about the triangle?

A što je s trokutom?

Well, I can rotate by a third of a turn clockwise

Evo, mogu ga zarotirati za trećinu okreta u smjeru kazaljke na satu

or a third of a turn anticlockwise.

ili za trećinu u suprotnom smjeru.

But now this has some reflectional symmetry.

Ali sada je tu refleksivna simetrija.

I can reflect it in the line through X,

Mogu ga reflektirati kroz točku X,

or the line through Y,

ili kroz točku Y,

or the line through Z.

ili kroz točku Z.

Five symmetries and then of course the zeroth symmetry

Pet simetrija i onda, naravno, nulta simetrija,

where I just pick it up and leave it where it is.

gdje ga podignem i spustim kako je bio.

So both of these objects have six symmetries.

Dakle, oba ova predmeta imaju šest simetrija.

Now, I'm a great believer that mathematics is not a spectator sport,

Ipak, uvjeren sam da matematika nije sport za gledatelje

and you have to do some mathematics

i da je morate malo vježbati

in order to really understand it.

kako biste je razumjeli.

So here is a little question for you.

Zato vam postavljam malo pitanje.

And I'm going to give a prize at the end of my talk

I nudim nagradu na kraju ovog predavanja

for the person who gets closest to the answer.

osobi koja dođe najbliže odgovoru.

The Rubik's Cube.

Rubikova kocka.

How many symmetries does a Rubik's Cube have?

Koliko simetrija ima Rubikova kocka?

How many things can I do to this object

Koliko joj stvari mogu učiniti

and put it down so it still looks like a cube?

i spustiti je da još uvijek izgleda kao kocka?

Okay? So I want you to think about that problem as we go on,

Želio bih da o tome razmišljate u nastavku

and count how many symmetries there are.

i brojite koliko simetrija ima.

And there will be a prize for the person who gets closest at the end.

A na kraju, nagrada čeka onoga koji bude najbliže odgovoru.

But let's go back down to symmetries that I got for these two objects.

No, vratimo se simetrijama koje sam dobio za ona dva predmeta.

What Galois realized: it isn't just the individual symmetries,

Galois je shvatio da se ne radi samo o pojedinačnim simetrijama,

but how they interact with each other

već kako one međusobno djeluju,

which really characterizes the symmetry of an object.

što stvarno određuje simetriju predmeta.

If I do one magic trick move followed by another,

Izvedem li jedan mađioničarski trik za drugim,

the combination is a third magic trick move.

kombinacija daje treći trik.

And here we see Galois starting to develop

I ovdje vidimo kako Galois počinje razvijati

a language to see the substance

jezik kojim bi vidio tvar

of the things unseen, the sort of abstract idea

nevidljivih stvari, nešto poput apstraktne ideje

of the symmetry underlying this physical object.

simetrije koja je temeljni dio tog fizičkog objekta.

For example, what if I turn the starfish

Primjerice, što ako zakrenem morsku zvijezdu

by a sixth of a turn,

za šestinu okreta

and then a third of a turn?

i potom za trećinu okreta?

So I've given names. The capital letters, A, B, C, D, E, F,

Stoga sam dao imena. Velika slova A, B, C, D, E, F

are the names for the rotations.

su imena rotacija.

B, for example, rotates the little yellow dot

B, primjerice, zakreće žutu točkicu

to the B on the starfish. And so on.

na B kod zvijezde. I tako dalje.

Dakle, što se zbiva okrenem li B, što je šestina okreta,

So what if I do B, which is a sixth of a turn,

followed by C, which is a third of a turn?

potom za C, što je trećina okreta?

Well let's do that. A sixth of a turn,

Idemo. Za šestinu okreta,

followed by a third of a turn,

potom za trećinu okreta,

the combined effect is as if I had just rotated it by half a turn in one go.

zbrojeni učinak je isti kao da sam je zarotirao za pola okreta odjednom.

So the little table here records

Ova tablica ovdje bilježi

how the algebra of these symmetries work.

kako funkcionira algebra simetrije.

I do one followed by another, the answer is

Jedan okret za drugim i odgovor je

it's rotation D, half a turn.

rotacija D, pola okreta.

What I if I did it in the other order? Would it make any difference?

Promijenim li redoslijed, bi li bilo ikakve razlike?

Let's see. Let's do the third of the turn first, and then the sixth of a turn.

Pogledajmo. Prvo trećinu okreta, potom šestinu okreta.

Of course, it doesn't make any difference.

Naravno, nema nikakve razlike.

It still ends up at half a turn.

I dalje ispada pola okreta.

And there is some symmetry here in the way the symmetries interact with each other.

I ovdje ima simetrije na način da simetrije međusobno djeluju.

But this is completely different to the symmetries of the triangle.

Ali to je potpuno drugačije od simetrija trokuta.

Let's see what happens if we do two symmetries

Pogledajmo što se događa ako su dvije simetrije

with the triangle, one after the other.

na trokutu, jedna nakon druge.

Let's do a rotation by a third of a turn anticlockwise,

Rotiramo za trećinu okreta u smjeru obrnuto od kazaljki na satu

and reflect in the line through X.

i reflektiramo kroz točku X.

Well, the combined effect is as if I had just done the reflection in the line through Z

Ovdje je ukupni efekt kao da smo izveli refleksiju kroz točku Z

to start with.

za početak.

Now, let's do it in a different order.

Učinimo to sada obrnutim redom.

Let's do the reflection in X first,

Prvo napravimo refleksiju kroz X,

followed by the rotation by a third of a turn anticlockwise.

slijedimo rotacijom za trećinu okreta suprotno od kretanja kazaljki na satu.

The combined effect, the triangle ends up somewhere completely different.

Ukupni učinak, trokut završava na sasvim drugom mjestu.

It's as if it was reflected in the line through Y.

Kao da je refleksija bila kroz točku Y.

Now it matters what order you do the operations in.

Sad je važno kojim redoslijedom obavljamo operacije.

And this enables us to distinguish

To nam omogućuje da razlučimo

why the symmetries of these objects --

zašto su simetrije ovih objekata -

they both have six symmetries. So why shouldn't we say

obje imaju po šest simetrija. Zašto ne bismo mogli reći

they have the same symmetries?

da imaju iste simetrije?

But the way the symmetries interact

Ali način na koji simetrije međusobno djeluju,

enable us -- we've now got a language

omogućuje nam, sada kada imamo jezik,

to distinguish why these symmetries are fundamentally different.

razlikovati zašto su te dvije simetrije u osnovi različite.

And you can try this when you go down to the pub, later on.

Ovo možete isprobati u kafiću, kasnije.

Take a beer mat and rotate it by a quarter of a turn,

Uzmite podložak za pivo i zakrenite ga za četvrtinu okreta,

then flip it. And then do it in the other order,

potom ga prevrnite, napravite drugim redoslijedom

and the picture will be facing in the opposite direction.

i slika će biti okrenuta u suprotnom smjeru.

Now, Galois produced some laws for how these tables -- how symmetries interact.

Galois je napravio zakone kako te tablice, te simetrije, međusobno djeluju.

It's almost like little Sudoku tables.

Gotovo su nalik Sudoku tablicama.

You don't see any symmetry twice

Ne vidite ni jednu simetriju dva puta

in any row or column.

ni u kojem redu ili stupcu.

And, using those rules, he was able to say

Koristeći se ovim pravilima, mogao je reći

da, zapravo, postoje samo dva predmeta

that there are in fact only two objects

with six symmetries.

sa šest simetrija.

And they'll be the same as the symmetries of the triangle,

I bit će poput simetrija trokuta,

or the symmetries of the six-pointed starfish.

ili poput simetrija šestokrake morske zvijezde.

I think this is an amazing development.

Zadivljujuć je to razvoj.

It's almost like the concept of number being developed for symmetry.

Gotovo kao da je koncept broja razvijen za simetriju.

In the front here, I've got one, two, three people

Ovdje u prvom redu, sjedi jedna, dvije, tri osobe

sitting on one, two, three chairs.

na jednoj, dvije, tri stolice.

The people and the chairs are very different,

Ljudi i stolice su vrlo različiti,

but the number, the abstract idea of the number, is the same.

ali broj, apstraktna ideja broja, je ista.

And we can see this now: we go back to the walls in the Alhambra.

I sada vidimo ovo: vratimo se zidovima Alhambre.

Here are two very different walls,

Ovdje su dva vrlo različita zida,

very different geometric pictures.

vrlo različitih geometrijskih slika.

But, using the language of Galois,

No, koristeći Galoisov jezik,

we can understand that the underlying abstract symmetries of these things

možemo razumjeti da je temeljna apstraktna simetrija ovih predmeta

are actually the same.

zapravo, jednaka.

For example, let's take this beautiful wall

Primjerice, ovaj prekrasan zid

with the triangles with a little twist on them.

s trokutima koji su malo zavinuti.

You can rotate them by a sixth of a turn

Možete ih zarotirati za šestinu okreta,

if you ignore the colors. We're not matching up the colors.

zanemarite li boje. Nećemo usklađivati boje,

But the shapes match up if I rotate by a sixth of a turn

ali oblici se slažu rotiram li za šestinu okreta

around the point where all the triangles meet.

oko točke gdje se svi trokuti sastaju.

What about the center of a triangle? I can rotate

Što je sa sredinom trokuta? Mogu rotirati

by a third of a turn around the center of the triangle,

za trećinu okreta oko centra trokuta

and everything matches up.

i sve dobro pristaje.

And then there is an interesting place halfway along an edge,

A tu je zatim zanimljivo mjesto na pola puta niz rub,

where I can rotate by 180 degrees.

gdje mogu zarotirati za 180 stupnjeva.

And all the tiles match up again.

I sve pločice opet pristaju.

So rotate along halfway along the edge, and they all match up.

Rotiran na pola puta po rubu i opet pristaju.

Now, let's move to the very different-looking wall in the Alhambra.

Prijeđimo sada do zida Alhambre koji potpuno drugačije izgleda.

And we find the same symmetries here, and the same interaction.

Vidimo ovdje iste simetrije i istu interakciju.

So, there was a sixth of a turn. A third of a turn where the Z pieces meet.

Okret za šestinu. Okret za trećinu gdje se sastaju Z dijelovi.

And the half a turn is halfway between the six pointed stars.

Pola okreta je na pola puta između šestokrakih zvijezda.

And although these walls look very different,

I premda ti zidovi izgledaju različito,

Galois has produced a language to say

Galois je stvorio jezik kojim kaže

that in fact the symmetries underlying these are exactly the same.

da su simetrije koje su temelji ovih, zapravo, potpuno jednake.

And it's a symmetry we call 6-3-2.

I ta se simetrija zove 6-3-2.

Here is another example in the Alhambra.

Evo još jednog primjera iz Alhambre.

This is a wall, a ceiling, and a floor.

Ovo su zid, strop i pod.

They all look very different. But this language allows us to say

Izgledaju vrlo različito, ali nam ovaj jezik omogućuje reći

that they are representations of the same symmetrical abstract object,

da oni predstavljaju isti simetrični apstraktni predmet

which we call 4-4-2. Nothing to do with football,

koji nazivamo 4-4-2. Nemaju veze s nogometom,

but because of the fact that there are two places where you can rotate

već zbog činjenice da postoje dva mjesta gdje možete rotirati

by a quarter of a turn, and one by half a turn.

za četvrt okreta i jedan za pola okreta.

Now, this power of the language is even more,

Moć ovog jezika je još snažnija

because Galois can say,

jer Galois može reći,

"Did the Moorish artists discover all of the possible symmetries

"Jesu li Maori otkrili sve moguće simetrije

on the walls in the Alhambra?"

na zidovima Alhambre?"

And it turns out they almost did.

I čini se da gotovo jesu.

You can prove, using Galois' language,

Možete dokazati, koristeći Galoisov 'jezik',

there are actually only 17

da postoji samo 17 različitih simetrija

different symmetries that you can do in the walls in the Alhambra.

koje možete izvesti na zidovima Alhambre.

And they, if you try to produce a different wall with this 18th one,

I pokušate li narisati novi zid pomoću osamnaeste simetrije,

it will have to have the same symmetries as one of these 17.

imat će iste simetrije kao jedan od ovih 17.

But these are things that we can see.

To su stvari koje možemo vidjeti.

And the power of Galois' mathematical language

I moć Galoisovog matematičkog jezika je u tome

is it also allows us to create

što nam omoguje stvoriti

symmetrical objects in the unseen world,

simetrične predmete u neviđenom svijetu,

beyond the two-dimensional, three-dimensional,

izvan dvodimenzionalnog, trodimenzionalnog,

all the way through to the four- or five- or infinite-dimensional space.

skroz do četiri- ili pet- ili beskonačno dimenzionalnog prostora.

And that's where I work. I create

I to je gdje ja radim. Stvaram

mathematical objects, symmetrical objects,

matematičke objekte, simetrične objekte,

using Galois' language,

koristeći Galoisov jezik,

in very high dimensional spaces.

u visoko dimenzioniranim prostorima.

So I think it's a great example of things unseen,

To je odličan primjer neviđenih stvari

which the power of mathematical language allows you to create.

koje vam moć matematičkog jezika dozvoljava stvoriti.

So, like Galois, I stayed up all last night

Tako sam, poput Galoisa, ostao prošlu noć budan,

creating a new mathematical symmetrical object for you,

kreirajući za vas jedan novi matematički simetrični predmet

and I've got a picture of it here.

i ovdje imam njegovu sliku.

Well, unfortunately it isn't really a picture. If I could have my board

No, nažalost, nije to stvarno slika. Mogu li dobiti svoju ploču

at the side here, great, excellent.

ovdje - odlično.

Here we are. Unfortunately, I can't show you

Evo nas. Nažalost, ne mogu vam pokazati

a picture of this symmetrical object.

sliku ovog simetričnog predmeta.

But here is the language which describes

Ali postoji jezik koji opisuje

how the symmetries interact.

kako njegove simetrije međusobno djeluju.

Now, this new symmetrical object

Ovaj novi simetričan predmet

does not have a name yet.

još nema ime.

Now, people like getting their names on things,

A ljudi vole da se po njima imenuju stvari,

on craters on the moon

krateri na Mjesecu

or new species of animals.

ili nove vrste životinja.

So I'm going to give you the chance to get your name on a new symmetrical object

Zato vam dajem priliku da date svoje ime ovom novom simetričnom objektu

which hasn't been named before.

koji još nema ime.

And this thing -- species die away,

A ova stvar - vrste izumiru,

and moons kind of get hit by meteors and explode --

mjesece pogađaju meteori i razlete se -

but this mathematical object will live forever.

ali ovaj matematički predmet će živjeti zauvijek.

It will make you immortal.

Učinit će vas besmrtnim.

In order to win this symmetrical object,

Kako biste osvojili ovaj simetrični predmet,

what you have to do is to answer the question I asked you at the beginning.

morate odgovoriti na pitanje koje sam postavio na početku.

How many symmetries does a Rubik's Cube have?

Koliko simetrija ima Rubikova kocka?

Okay, I'm going to sort you out.

Evo, pomoći ću vam.

Rather than you all shouting out, I want you to count how many digits there are

Umjesto da svi izvikujete, želim da prebrojite koliko znamenki

ima u tom broju, dobro?

in that number. Okay?

If you've got it as a factorial, you've got to expand the factorials.

Ako imate faktorijel (umnožak brojeva), morate faktorijele proširiti.

Okay, now if you want to play,

OK, ako želite sudjelovati,

I want you to stand up, okay?

želim da ustanete, dobro?

Mislite li da imate procjenu broja znamenki,

If you think you've got an estimate for how many digits,

right -- we've already got one competitor here.

dobro - imamo jednog natjecatelja ovdje.

If you all stay down he wins it automatically.

Ostanete li svi sjediti, on automatski pobjeđuje.

Okay. Excellent. So we've got four here, five, six.

Ok. Odlično. Ovdje vas je četiri, pet, šest.

Great. Excellent. That should get us going. All right.

Odlično. Sada možemo krenuti.

Anybody with five or less digits, you've got to sit down,

Svi koji imate broj od 5 ili manje znamenki, možete sjesti

because you've underestimated.

jer je procjena preniska.

Five or less digits. So, if you're in the tens of thousands you've got to sit down.

Pet ili manje znamenki. Ako imate desetke tisuća, morate sjesti.

60 digits or more, you've got to sit down.

60 znamenki ili više, morate sjesti.

You've overestimated.

Procjena je previsoka.

20 digits or less, sit down.

20 znamenki ili manje, sjednite.

How many digits are there in your number?

Koliko znamenki ima vaš broj?

Two? So you should have sat down earlier.

Dvije? Trebali ste već ranije sjesti.

(Laughter)

(Smijeh)

Let's have the other ones, who sat down during the 20, up again. Okay?

Pogledajmo ostale. Tko je sjeo kod 20, neka ustane.

If I told you 20 or less, stand up.

Ako sam rekao 20 ili manje, ustanite.

Because this one. I think there were a few here.

Zbog ovoga. Mislim da ih je ovdje bilo nekoliko.

The people who just last sat down.

Ljudi koji su upravo sjeli.

Okay, how many digits do you have in your number?

OK, koliko znamenki imate u svom broju?

(Laughs)

(Smijeh)

21. Okay good. How many do you have in yours?

21. OK. Dobro. Koliko imate vi u svom?

18. So it goes to this lady here.

18. Dakle, ide ovoj dami ovdje.

21 is the closest.

21 je najbliže.

It actually has -- the number of symmetries in the Rubik's cube

Zapravo ima - broj simetrija u Rubikovoj kocki

has 25 digits.

ima 25 znamenki.

So now I need to name this object.

Sada trebam imenovati ovaj predmet.

So, what is your name?

Dakle, kako se zovete?

I need your surname. Symmetrical objects generally --

Treba mi Vaše prezime. Simetrični predmeti najčešće -

spell it for me.

sričite mi svoje ime.

G-H-E-Z

G-H-E-Z

No, SO2 has already been used, actually,

Ne, SO2 je već iskorišteno

in the mathematical language. So you can't have that one.

u matematičkom jeziku pa ne može ići ovdje.

So Ghez, there we go. That's your new symmetrical object.

Znači Ghez, eto ga. Ovo je Vaš novi simetrični predmet.

You are now immortal.

Postali ste besmrtni.

(Pljesak)

(Applause)

And if you'd like your own symmetrical object,

Želite li svoj vlastiti simetrični predmet,

I have a project raising money for a charity in Guatemala,

imam dobrotvorni projekt prikupljanja sredstava u Gvatemali,

where I will stay up all night and devise an object for you,

gdje ću cijelu noć probdjeti i osmišljavati predmet za Vas,

for a donation to this charity to help kids get into education in Guatemala.

za donaciju dobrotvornoj organizaciji koja se zalaže za obrazovanje djece Gvatemale.

And I think what drives me, as a mathematician,

I mislim da ono što me pokreće kao matematičara,

are those things which are not seen, the things that we haven't discovered.

su te stvari koje se ne vide, stvari koje još nismo otkrili.

It's all the unanswered questions which make mathematics a living subject.

Sva ta neodgovorena pitanja koja čine matematiku živom.

And I will always come back to this quote from the Japanese "Essays in Idleness":

I uvijek ću se vraćati tom citatu iz japanskih 'Eseja iz dosade':

"In everything, uniformity is undesirable.

"U svemu je jednolikost nepoželjna.

Leaving something incomplete makes it interesting,

Ostavljajući nešto nedovršenim, to ga čini zanimljivim

and gives one the feeling that there is room for growth." Thank you.

i daje osjećaj da ima mjesta za rast." Hvala vam.

(Applause)

(Pljesak)