On the 30th of May, 1832, a gunshot was heard ringing out across the 13th arrondissement in Paris. (Gunshot) A 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 Evariste 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 were, "Don't cry for me, Alfred. I need all the courage I can muster to die at the age of 20."
30. svibnja 1832. čuo se pucanj kako odzvanja u 13. pariškom okrugu. (Pucanj) Tog je jutra jedan seljak, na svom putu prema tržnici, potrčao prema mjestu pucnja i zatekao na podu mladića u agoniji, očigledno ustrijeljenog u dvoboju. Ime tog mladića bilo je Evariste Galois. Bio je poznati pariški revolucionar tog vremena. Galois je prebačen u mjesnu bolnicu, gdje umire sljedećeg dana u naručju svoga brata. Posljednje riječi koje je rekao svom bratu bile su, „Ne plači zbog mene, Alfrede. Potrebna mi je sva moguća snaga kako bih umro u dvadesetoj."
It 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. (Laughter) This is how he wrote most of his mathematics.
Zapravo, Galois se nije proslavio zbog neke revolucionarne politike. Nekoliko godina ranije, dok je još bio u školi, riješio je jedan od velikih matematičkih problema tog vremena. Pisao je pariškim akademicima, pokušavajući objasniti svoju teoriju. Ali akademici nisu razumjeli ništa od onoga što je napisao. (Smijeh) Ovako je pisao većinu svoje matematike.
So, the night before that duel, he realized 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 that he stayed up all night doing mathematics was the fact that he was such a bad shot that morning and got killed.
Noć prije dvoboja, shvatio je da mu je to možda zadnja prilika kako bi razjasnio svoje veliko otkriće. Ostao je budan cijelu noć, pišući i pokušavajući objasniti svoje ideje. Dolaskom zore otišao je ususret svojoj sudbini, ostavivši hrpu papira na svom stolu za buduću generaciju. Možda je to što je probdio noć rješavajući matematiku, doprinijelo tome da to jutro tako loše puca i pogine.
But contained inside those documents was a new language, a language to understand one of 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.
Ali, sadržaj u tim papirima bio je nov jezik, jezik za razumijevanje jednog od osnovnih koncepata znanosti -- a to je simetrija. Simetrija je, gotovo, jezik prirode. Pomaže nam shvatiti toliko različitih komadića znanstvenog svijeta. Primjerice, molekularnu strukturu. Kakvi kristali su mogući, možemo shvatiti kroz matematiku simetrije.
In microbiology you really don't want to get a symmetrical object, because they are 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 on a larger scale of biology, actually symmetry is very important, because it actually communicates genetic information.
U mikrobiologiji zaista ne želite dobiti simetričan objekt, jer su načelno dosta gadni. Virus svinjske gripe, trenutno, je simetričan. A koristi se učinkovitošću simetrije kako bi se tako dobro širio. I u širim razmjerima biologije simetrija je jako važna, jer, zapravo, prenosi genetske podatke.
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're probably drawn to the lower two. Because it is 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.
Snimio sam dvije slike a onda ih umjetno napravio simetričnima. Pitam li vas koje su vam ljepše, vjerojatno će vas privući donje dvije. To je zato jer je teško postići simetriju. Učinite li sebe simetričnima, šaljete signal da su vaši geni dobri, da ste dobro odgojeni, što navodi da ćete biti kvalitetan partner. Simetrija je, stoga, jezik koji vam pomaže poslati genetsku poruku.
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 they are all facets of some strange symmetrical shape in a higher dimensional space.
Simetrija nam isto pomaže objasniti što se događa u Velikom hadronskom sudaraču u CERN-u. Ili što se ne događa u Velikom hadronskom sudaraču u CERN-u. Kako bismo predvidjeli koje elementarne čestice bismo mogli tamo vidjeti, čini se da su to sve aspekti nekog čudnog simetričnog oblika u prostoru širih dimenzija.
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 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 geometric figures, without which means it is humanly impossible to comprehend a single word."
Mislim da je Galileo jako dobro izložio moć matematike u razumijevanju znanstvenog svijeta oko nas. Napisao je: „Svemir se ne može pročitati dok nismo naučili jezik i upoznali se s pismom kojim je napisan. Napisan je matematičkim jezikom, a slova su mu trokuti, krugovi i drugi geometrijski oblici, bez kojih je nemoguće shvatiti ijednu riječ."
But it's not just scientists who are interested in symmetry. Artists too love to play around with symmetry. They also have a slightly more ambiguous relationship with it. Here is 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, found it deathly, the very marrow of death."
Ali nisu samo znanstvenici zainteresirani za simetriju. Umjetnici se isto vole igrati simetrijom. Također imaju i dvosmisleniji odnos s njom. Ovdje Thomas Mann govori o simetriji u "Čudesnoj gori". Njegov lik objašnjava snježnu pahulju i kaže kako je "zadrhtao nad njenom savršenom preciznošću, smatra je smrtonosnom, poput same srži smrti."
But what artists 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 Kurokawa. 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 them. Seven of them are exactly the same, and the eighth one is turned upside down.
Umjetnici vole postaviti očekivanja simetrije i onda ih uništiti. Prekrasan primjer za to našao sam kad sam posjetio svog kolegu u Japanu, profesora Kurokawu. Odveo me je u hramove grada Nikko. Odmah nakon ove slike, uspinjali smo se stepenicama. I prolaz koji vidite iza nas ima osam stupova ukrašenih prekrasnim simetričnim oblicima. Sedam ih je potpuno jednakih, a osmi je preokrenut.
And I said to Professor Kurokawa, "Wow, the architects must have really been kicking themselves when they realized that they'd made a mistake and put this one upside down." And he said, "No, no, no. It was a very deliberate act." And he referred me to this lovely quote from the Japanese "Essays in Idleness" from the 14th century, in which 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." Even when building the Imperial Palace, they always leave one place unfinished.
Rekao sam profesoru Kurokawi, sigurno su se arhitekti lupali po glavi kada su shvatili da su pogriješili i ovoga preokrenuli. A on kaže: "Ne, ne, ne. To je hotimično napravljeno." I onda me uputio na prekrasni citat iz japanskih "Eseja o ljenčarenju" iz 14. stoljeća, u kojima pisac kaže, "U svemu je jednoličnost nepoželjna. Ostavljati nešto nepotpunim, čini ga zanimljivim i ostavlja dojam da ima prostora za rast." Čak i pri gradnji carske palače, uvijek su ostavili jedan dio nedovršen.
But if I had to choose one building in the world to be cast out on a 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 these rather kind of nerdy mathematical trips, which my family love. This is my son Tamer. 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 doesn't look at how mathematics is embedded in the world we live in. So, I wanted to open his eyes up to how much symmetry is running through the Alhambra.
Ali kad bih izabrao jednu zgradu na svijetu koju bih preselio na pusti otok gdje bih proživio ostatak života, kako sam ovisnik o simetriji, vjerojatno bih izabrao Alhambru u Granadi. Ova palača slavi simetriju. Nedavno sam odveo svoju obitelj -- odlazimo na ovakve štreberske matematičke izlete koje moja obitelj obožava. Evo mog sina Tamera. Vidite da se stvarno zabavlja na našem matematičkom putovanju u Alhambru. Želio sam obogatiti njegovo iskustvo. Smatram da je problem sa školskom matematikom u tome što ne promatra koliko je matematika utkana u svijet u kojem živimo. Zato sam želio da uoči koliko se simetrije nalazi u Alhambri.
You see it already. 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 were denied the possibility to draw things with souls. So they explored a more geometric art. And so what is symmetry? The Alhambra somehow asks all of these questions. What is symmetry? When [there] are two of these walls, do they have the same symmetries? Can we say whether they discovered all of the symmetries in the Alhambra?
Vidi se odmah. Čim uđete, vidite reflektivnu simetriju vode. No, na zidovima se nalazi ono najuzbudljivije. Maurskim umjetnicima nije bilo dozvoljeno crtati ono što ima dušu. Stoga su istraživali geometrijsku umjetnost. Dakle, što je simetrija? Alhambra, zapravo, postavlja sva ta pitanja. Što je simetrija? Ako imamo dva ovakva zida, jesu li oni jednako simetrični? Možemo li reći da su otkrili svu simetriju u Alhambri?
And it was Galois who produced a language to be able to answer some of these questions. For Galois, symmetry -- unlike for Thomas Mann, which was something still and deathly -- for 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. It looks like it did before it started.
Galois je taj koji je stvorio jezik kako bi odgovorio na neka od tih pitanja. Za Galoisa je simetrija, za razliku od Thomasa Manna, kome je ona nepomična i samrtnička -- za Galoisu je simetrija stvar gibanja. Što možete učiniti simetričnom objektu, pomaknete li ga nekamo, da izgleda isti kao prije pomicanja? Volim to opisati kao pokrete mađioničarskog trika. Što možete nečemu učiniti? Zatvorite oči. Nešto učinim i spustim. Izgleda kao na početku.
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'd 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.
Tako, primjerice, zidovi Alhambre -- Mogu uzeti sve te pločice i pričvrstiti ih na žutom dijelu, zarotirati ih za 90 stupnjeva, spustiti ih natrag i savršeno odgovaraju. I kad biste ponovo otvorili oči, ne biste znali da su pomaknute. Ali, upravo je pomak taj koji karakterizira simetriju unutar Alhambre. A radi se i o stvaranju jezika kojim bi se to opisalo. Moć matematike je često 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 -- 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 starfish. What can I do to the 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 could rotate it by a third of a turn, or a half a turn, or put it back down on its image, or two thirds of a turn. And a fifth symmetry, I can rotate it by five sixths of a turn. And those are things that I can do to the symmetrical object that make it look like it did before I started.
Stoga ću vas provesti, možda malo pogurnuti matematički pa se pripremite -- pogurati vas malo kako biste razumjeli kako radi ovaj jezik, koji nam omogućuje usvojiti što je simetrija. Stoga, uzmimo ova dva simetrična predmeta. Uzmimo uvijenu šesterokraku morsku zvijezdu. Što mogu učiniti morskoj zvijezdi da bi izgledala isto? Evo, rotirao sam je za šestinu okreta i dalje izgleda kao na početku. Mogu je zakrenuti za trećinu okreta, ili pola okreta, ili spustiti na njezin odraz, ili za dvije trećine okreta. I peta simetrija, mogu je zarotirati za pet šestina okreta. To su stvari koje mogu učiniti simetričnom predmetu, a da izgleda kao na početku.
Now, for Galois, there was actually a sixth symmetry. Can anybody think what else I could do to this which would leave it like I did before I started? I can't flip it because I've put a little twist on it, haven't I? 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 the number zero was a very modern concept, seventh century A.D., by the Indians. It seems mad to talk about nothing. And this is the same idea. This is a symmetrical -- so everything has symmetry, where you just leave it where it is.
Za Galoisa je postojala i šesta simetrija. Može li se itko sjetiti što bih još mogao učiniti, a da bude kao na početku? Ne mogu je preokrenuti jer sam stavio oznaku, zar ne? Nema refleksivnu simetriju. Ali bih je mogao ostaviti ovdje gdje jest, podignuti i ponovno spustiti. Galoisu je to bilo poput nulte simetrije. Zapravo, izum broja nula zapravo je moderan koncept Indijaca iz sedmog stoljeća p. K. Čini se ludim govoriti o ničemu. A ovo je isto to. To je simetrično -- stoga sve ima simetriju, samo to ostavite gdje jest.
So, this object has six symmetries. And what about the triangle? Well, I can rotate by a third of a turn clockwise or a third of a turn anticlockwise. 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 zeroth 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.
Dakle, ovaj predmet ima šest simetrija. A što je s trokutom? Evo, mogu ga zarotirati za trećinu okreta u smjeru kazaljke na satu ili za trećinu u suprotnom smjeru. Ali sada je tu refleksivna simetrija. Mogu ga reflektirati kroz točku X, ili kroz točku Y, ili kroz točku Z. Pet simetrija i onda, naravno, nulta simetrija, gdje ga podignem i spustim kako je bio. Dakle, oba ova predmeta imaju šest simetrija. Ipak, uvjeren sam da matematika nije sport za gledatelje i da je morate malo vježbati kako biste je razumjeli.
So here is a little question for you. And I'm going to give 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 will be a prize for the person who gets closest at the end.
Zato vam postavljam malo pitanje. I nudim nagradu na kraju ovog predavanja osobi koja dođe najbliže odgovoru. Rubikova kocka. Koliko simetrija ima Rubikova kocka? Koliko joj stvari mogu učiniti i spustiti je da još uvijek izgleda kao kocka? Želio bih da o tome razmišljate u nastavku i brojite koliko simetrija ima. 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. What Galois realized: 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 if I turn the starfish by a sixth of a turn, and then a third of a turn?
No, vratimo se simetrijama koje sam dobio za ona dva predmeta. Galois je shvatio da se ne radi samo o pojedinačnim simetrijama, već kako one međusobno djeluju, što stvarno određuje simetriju predmeta. Izvedem li jedan mađioničarski trik za drugim, kombinacija daje treći trik. I ovdje vidimo kako Galois počinje razvijati jezik kojim bi vidio tvar nevidljivih stvari, nešto poput apstraktne ideje simetrije koja je temeljni dio tog fizičkog objekta. Primjerice, što ako zakrenem morsku zvijezdu za šestinu okreta i potom za trećinu okreta?
So I've given names. The capital letters, A, B, C, D, E, F, are the names for the rotations. B, for example, rotates the little yellow dot to the B 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 is as if I had 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 it's rotation D, half a turn. What I if I did it in the other order? Would it make any difference? 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.
Stoga sam dao imena. Velika slova A, B, C, D, E, F su imena rotacija. B, primjerice, zakreće žutu točkicu na B kod zvijezde. I tako dalje. Dakle, što se zbiva okrenem li B, što je šestina okreta, potom za C, što je trećina okreta? Idemo. Za šestinu okreta, potom za trećinu okreta, zbrojeni učinak je isti kao da sam je zarotirao za pola okreta odjednom. Ova tablica ovdje bilježi kako funkcionira algebra simetrije. Jedan okret za drugim i odgovor je rotacija D, pola okreta. Promijenim li redoslijed, bi li bilo ikakve razlike? Pogledajmo. Prvo trećinu okreta, potom šestinu okreta. Naravno, nema nikakve razlike. I dalje ispada pola okreta.
And there is 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 do two symmetries with the triangle, one after the other. Let's do a rotation by a third of a turn anticlockwise, and reflect in the line through X. Well, the combined effect is as if I had 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 anticlockwise. The combined effect, the triangle ends up somewhere completely different. It's as if it was reflected in the line through Y.
I ovdje ima simetrije na način da simetrije međusobno djeluju. Ali to je potpuno drugačije od simetrija trokuta. Pogledajmo što se događa ako su dvije simetrije na trokutu, jedna nakon druge. Rotiramo za trećinu okreta u smjeru obrnuto od kazaljki na satu i reflektiramo kroz točku X. Ovdje je ukupni efekt kao da smo izveli refleksiju kroz točku Z za početak. Učinimo to sada obrnutim redom. Prvo napravimo refleksiju kroz X, slijedimo rotacijom za trećinu okreta suprotno od kretanja kazaljki na satu. Ukupni učinak, trokut završava na sasvim drugom mjestu. Kao da je refleksija bila kroz točku Y.
Now it matters what order you do the operations in. And this enables 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 to distinguish why these symmetries are fundamentally different. And you can try this when you go down to the pub, later on. Take a beer mat and rotate it by a 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.
Sad je važno kojim redoslijedom obavljamo operacije. To nam omogućuje da razlučimo zašto su simetrije ovih objekata - obje imaju po šest simetrija. Zašto ne bismo mogli reći da imaju iste simetrije? Ali način na koji simetrije međusobno djeluju, omogućuje nam, sada kada imamo jezik, razlikovati zašto su te dvije simetrije u osnovi različite. Ovo možete isprobati u kafiću, kasnije. Uzmite podložak za pivo i zakrenite ga za četvrtinu okreta, potom ga prevrnite, napravite drugim redoslijedom i slika će biti okrenuta u suprotnom smjeru.
Now, Galois produced some laws for how these tables -- how symmetries interact. It's almost like little Sudoku tables. You don't see any symmetry twice in 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 starfish. I think this is an amazing development. It's almost like the concept of number being developed for symmetry. In the front here, 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.
Galois je napravio zakone kako te tablice, te simetrije, međusobno djeluju. Gotovo su nalik Sudoku tablicama. Ne vidite ni jednu simetriju dva puta ni u kojem redu ili stupcu. Koristeći se ovim pravilima, mogao je reći da, zapravo, postoje samo dva predmeta sa šest simetrija. I bit će poput simetrija trokuta, ili poput simetrija šestokrake morske zvijezde. Zadivljujuć je to razvoj. Gotovo kao da je koncept broja razvijen za simetriju. Ovdje u prvom redu, sjedi jedna, dvije, tri osobe na jednoj, dvije, tri stolice. Ljudi i stolice su vrlo različiti, ali broj, apstraktna ideja broja, je ista.
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 with 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 a sixth of a turn around the point where all the triangles meet. What about the center of a triangle? I can rotate by a third of a turn around the center of the triangle, and everything matches up. And then there is an interesting place halfway along an edge, where I can rotate by 180 degrees. And all the tiles match up again. So rotate along halfway along the edge, and they all match up.
I sada vidimo ovo: vratimo se zidovima Alhambre. Ovdje su dva vrlo različita zida, vrlo različitih geometrijskih slika. No, koristeći Galoisov jezik, možemo razumjeti da je temeljna apstraktna simetrija ovih predmeta zapravo, jednaka. Primjerice, ovaj prekrasan zid s trokutima koji su malo zavinuti. Možete ih zarotirati za šestinu okreta, zanemarite li boje. Nećemo usklađivati boje, ali oblici se slažu rotiram li za šestinu okreta oko točke gdje se svi trokuti sastaju. Što je sa sredinom trokuta? Mogu rotirati za trećinu okreta oko centra trokuta i sve dobro pristaje. A tu je zatim zanimljivo mjesto na pola puta niz rub, gdje mogu zarotirati za 180 stupnjeva. I sve pločice opet pristaju. Rotiran na pola puta po rubu i opet pristaju.
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 a turn. A third of a turn where the Z pieces meet. And 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 symmetries underlying these are exactly the same. And it's a symmetry we call 6-3-2.
Prijeđimo sada do zida Alhambre koji potpuno drugačije izgleda. Vidimo ovdje iste simetrije i istu interakciju. Okret za šestinu. Okret za trećinu gdje se sastaju Z dijelovi. Pola okreta je na pola puta između šestokrakih zvijezda. I premda ti zidovi izgledaju različito, Galois je stvorio jezik kojim kaže da su simetrije koje su temelji ovih, zapravo, potpuno jednake. I ta se simetrija zove 6-3-2.
Here is 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 that they are representations of the 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.
Evo još jednog primjera iz Alhambre. Ovo su zid, strop i pod. Izgledaju vrlo različito, ali nam ovaj jezik omogućuje reći da oni predstavljaju isti simetrični apstraktni predmet koji nazivamo 4-4-2. Nemaju veze s nogometom, već zbog činjenice da postoje dva mjesta gdje možete rotirati za četvrt okreta i jedan za pola okreta.
Now, this power 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 to produce a different wall with this 18th one, it will have to have the same symmetries as one of these 17.
Moć ovog jezika je još snažnija jer Galois može reći, "Jesu li Maori otkrili sve moguće simetrije na zidovima Alhambre?" I čini se da gotovo jesu. Možete dokazati, koristeći Galoisov 'jezik', da postoji samo 17 različitih simetrija koje možete izvesti na zidovima Alhambre. I pokušate li narisati novi zid pomoću osamnaeste simetrije, imat će iste simetrije kao jedan od ovih 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 four- or five- or infinite-dimensional space. And that's where I work. I create mathematical objects, symmetrical objects, using Galois' 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.
To su stvari koje možemo vidjeti. I moć Galoisovog matematičkog jezika je u tome što nam omoguje stvoriti simetrične predmete u neviđenom svijetu, izvan dvodimenzionalnog, trodimenzionalnog, skroz do četiri- ili pet- ili beskonačno dimenzionalnog prostora. I to je gdje ja radim. Stvaram matematičke objekte, simetrične objekte, koristeći Galoisov jezik, u visoko dimenzioniranim prostorima. To je odličan primjer neviđenih stvari koje vam moć matematičkog jezika dozvoljava stvoriti.
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, unfortunately it isn't really a picture. If I could have my board at the side here, great, excellent. Here we are. Unfortunately, I can't show you a picture of this symmetrical object. But here is the language which describes how the symmetries interact.
Tako sam, poput Galoisa, ostao prošlu noć budan, kreirajući za vas jedan novi matematički simetrični predmet i ovdje imam njegovu sliku. No, nažalost, nije to stvarno slika. Mogu li dobiti svoju ploču ovdje - odlično. Evo nas. Nažalost, ne mogu vam pokazati sliku ovog simetričnog predmeta. Ali postoji jezik koji opisuje kako njegove simetrije međusobno djeluju.
Now, this new symmetrical object does not have a name yet. Now, people like getting their names on things, on 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 die 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 this symmetrical object, what you have to do is to answer the question I asked you at the beginning. How many symmetries does a Rubik's Cube have?
Ovaj novi simetričan predmet još nema ime. A ljudi vole da se po njima imenuju stvari, krateri na Mjesecu ili nove vrste životinja. Zato vam dajem priliku da date svoje ime ovom novom simetričnom objektu koji još nema ime. A ova stvar - vrste izumiru, mjesece pogađaju meteori i razlete se - ali ovaj matematički predmet će živjeti zauvijek. Učinit će vas besmrtnim. Kako biste osvojili ovaj simetrični predmet, morate odgovoriti na pitanje koje sam postavio na početku. Koliko simetrija ima Rubikova kocka?
Okay, 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've got to expand the factorials. Okay, now if you want to play, I want you to stand up, okay? If you think you've got an estimate for how many digits, right -- we've already got one competitor here. If you all stay down he wins it automatically. Okay. Excellent. So we've got four here, five, six. Great. Excellent. That should get us going. All right.
Evo, pomoći ću vam. Umjesto da svi izvikujete, želim da prebrojite koliko znamenki ima u tom broju, dobro? Ako imate faktorijel (umnožak brojeva), morate faktorijele proširiti. OK, ako želite sudjelovati, želim da ustanete, dobro? Mislite li da imate procjenu broja znamenki, dobro - imamo jednog natjecatelja ovdje. Ostanete li svi sjediti, on automatski pobjeđuje. Ok. Odlično. Ovdje vas je četiri, pet, šest. Odlično. Sada možemo krenuti.
Anybody with five or less digits, you've got to sit down, because you've underestimated. Five or less digits. So, if you're in the tens 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. How many digits are there in your number? Two? So you should have sat down earlier. (Laughter) Let's have the other ones, who sat down during the 20, up again. Okay? If I told you 20 or less, stand up. Because this one. I think there were a few here. The people who just last sat down.
Svi koji imate broj od 5 ili manje znamenki, možete sjesti jer je procjena preniska. Pet ili manje znamenki. Ako imate desetke tisuća, morate sjesti. 60 znamenki ili više, morate sjesti. Procjena je previsoka. 20 znamenki ili manje, sjednite. Koliko znamenki ima vaš broj? Dvije? Trebali ste već ranije sjesti. (Smijeh) Pogledajmo ostale. Tko je sjeo kod 20, neka ustane. Ako sam rekao 20 ili manje, ustanite. Zbog ovoga. Mislim da ih je ovdje bilo nekoliko. Ljudi koji su upravo sjeli.
Okay, how many digits do you have in your number? (Laughs) 21. Okay good. How many do you have in yours? 18. So it goes to this lady here. 21 is the closest. It 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. Symmetrical objects generally -- spell it for me. G-H-E-Z No, SO2 has already been used, actually, in the mathematical language. So you can't have that one. So Ghez, there we go. That's your new symmetrical object. You are now immortal. (Applause)
OK, koliko znamenki imate u svom broju? (Smijeh) 21. OK. Dobro. Koliko imate vi u svom? 18. Dakle, ide ovoj dami ovdje. 21 je najbliže. Zapravo ima - broj simetrija u Rubikovoj kocki ima 25 znamenki. Sada trebam imenovati ovaj predmet. Dakle, kako se zovete? Treba mi Vaše prezime. Simetrični predmeti najčešće - sričite mi svoje ime. G-H-E-Z Ne, SO2 je već iskorišteno u matematičkom jeziku pa ne može ići ovdje. Znači Ghez, eto ga. Ovo je Vaš novi simetrični predmet. Postali ste besmrtni. (Pljesak)
And if you'd like your own symmetrical object, I have a project 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, as a mathematician, are those things which are not seen, the things that we haven't discovered. It's all the unanswered questions which make mathematics a living subject. And I will always come back to this quote from the Japanese "Essays in 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. (Applause)
Želite li svoj vlastiti simetrični predmet, imam dobrotvorni projekt prikupljanja sredstava u Gvatemali, gdje ću cijelu noć probdjeti i osmišljavati predmet za Vas, za donaciju dobrotvornoj organizaciji koja se zalaže za obrazovanje djece Gvatemale. I mislim da ono što me pokreće kao matematičara, su te stvari koje se ne vide, stvari koje još nismo otkrili. Sva ta neodgovorena pitanja koja čine matematiku živom. I uvijek ću se vraćati tom citatu iz japanskih 'Eseja iz dosade': "U svemu je jednolikost nepoželjna. Ostavljajući nešto nedovršenim, to ga čini zanimljivim i daje osjećaj da ima mjesta za rast." Hvala vam. (Pljesak)