Thank you very much. Please excuse me for sitting; I'm very old. (Laughter) Well, the topic I'm going to discuss is one which is, in a certain sense, very peculiar because it's very old. Roughness is part of human life forever and forever, and ancient authors have written about it. It was very much uncontrollable, and in a certain sense, it seemed to be the extreme of complexity, just a mess, a mess and a mess. There are many different kinds of mess. Now, in fact, by a complete fluke, I got involved many years ago in a study of this form of complexity, and to my utter amazement, I found traces -- very strong traces, I must say -- of order in that roughness. And so today, I would like to present to you a few examples of what this represents. I prefer the word roughness to the word irregularity because irregularity -- to someone who had Latin in my long-past youth -- means the contrary of regularity. But it is not so. Regularity is the contrary of roughness because the basic aspect of the world is very rough.
Puno vam hvala. Oprostite što sjedim; vrlo sam star. (Smijeh) Dakle, tema o kojoj ću govoriti je u određenom smislu, vrlo neobična jer je vrlo stara. Hrapavost je od pamtivijeka dio ljudskog života i drevni su pisci pisali o njoj. Nije ju bilo moguće kontrolirati, i na neki način, čini se krajnje složena, zbrkana i zbrčkana. Mnogo je vrsta zbrkanosti. Zapravo, pukim slučajem, pred mnogo godina, bio sam uključen u studiju tog oblika kompleksnosti, i na moje veliko iznenađenje, našao sam tragove -- vrlo jasne tragove, moram reći -- reda u toj hrapavosti. Tako vam danas želim dati nekoliko primjera onog što to predstavlja. Draža mi je riječ hrapavost nego riječ nepravilnost jer nepravilnost -- nekome tko je imao latinski u svojoj dalekoj mladosti -- znači suprotno od pravilnosti. Ali to nije tako. Pravilnost je suprotna hrapavosti jer je osnovni aspekt svijeta vrlo hrapav.
So let me show you a few objects. Some of them are artificial. Others of them are very real, in a certain sense. Now this is the real. It's a cauliflower. Now why do I show a cauliflower, a very ordinary and ancient vegetable? Because old and ancient as it may be, it's very complicated and it's very simple, both at the same time. If you try to weigh it -- of course it's very easy to weigh it, and when you eat it, the weight matters -- but suppose you try to measure its surface. Well, it's very interesting. If you cut, with a sharp knife, one of the florets of a cauliflower and look at it separately, you think of a whole cauliflower, but smaller. And then you cut again, again, again, again, again, again, again, again, again, and you still get small cauliflowers. So the experience of humanity has always been that there are some shapes which have this peculiar property, that each part is like the whole, but smaller. Now, what did humanity do with that? Very, very little. (Laughter)
Da vam pokažem nekoliko predmeta. Neki od njih su umjetni. Drugi od njih su vrlo stvarni, u nekom smislu. Ovo je stvarno. To je šenon. A zašto vam pokazujem šenon, tako obično i drevno povrće? Jer, koliko god da bilo staro i drevno, vrlo je komplicirano i vrlo jednostavno, sve u isti mah. Važete li ga -- naravno, lako ga je izvagati, a kad ga jedete, težina je važna -- no pretpostavimo da pokušate izmjeriti njegovu površinu. Dakle, vrlo je zanimljivo. Odrežete li oštrim nožem jednu od cvjetnih grana šenona i pogledate je zasebno, mislit ćete na cijeli šenon, ali manji. A onda opet odrežete i opet i opet i opet i opet i opet i još uvijek dobivate male šenone. Ljudskom je iskustvu dakle oduvijek poznato da postoje neki oblici s tim posebnim svojstvom, da je svaki dio poput cjeline, ali manji. A što je čovječanstvo s time napravilo? Vrlo, vrlo malo. (Smijeh)
So what I did actually is to study this problem, and I found something quite surprising. That one can measure roughness by a number, a number, 2.3, 1.2 and sometimes much more. One day, a friend of mine, to bug me, brought a picture and said, "What is the roughness of this curve?" I said, "Well, just short of 1.5." It was 1.48. Now, it didn't take me any time. I've been looking at these things for so long. So these numbers are the numbers which denote the roughness of these surfaces. I hasten to say that these surfaces are completely artificial. They were done on a computer, and the only input is a number, and that number is roughness. So on the left, I took the roughness copied from many landscapes. To the right, I took a higher roughness. So the eye, after a while, can distinguish these two very well.
I što sam zapravo učinio - proučavao sam ovaj problem i otkrio nešto prilično iznenađujuće. Da se hrapavost može mjeriti nekim brojem, 2,3; 1,2 a katkad i nekim većim. Jednom mi je neki prijatelj, da me zagnjavi, donio neku sliku i pitao, “Kolika je hrapavost ove krivulje?” Rekao sam, “Nešto manja od 1,5.” Bila je 1,48. Dakle, nije mi trebalo puno vremena. Dugo sam proučavao te stvari. To su dakle brojevi koji mogu opisati hrapavost tih površina. Moram vam reći da su te površine potpuno umjetne. Napravljene su računalom, a jedini je unos bio broj, a taj je broj hrapavost. Tako sam na lijevoj slici uzeo hrapavost kopiranu s mnogih pejzaža. Na desnoj sam uzeo veću hrapavost. Tako oko, nakon nekog vremena može dobro razlikovati jednu od druge.
Humanity had to learn about measuring roughness. This is very rough, and this is sort of smooth, and this perfectly smooth. Very few things are very smooth. So then if you try to ask questions: "What's the surface of a cauliflower?" Well, you measure and measure and measure. Each time you're closer, it gets bigger, down to very, very small distances. What's the length of the coastline of these lakes? The closer you measure, the longer it is. The concept of length of coastline, which seems to be so natural because it's given in many cases, is, in fact, complete fallacy; there's no such thing. You must do it differently.
Čovječanstvo je moralo učiti mjeriti hrapavost. Ovo je vrlo hrapavo, ovo donekle glatko, a ovo je savršeno glatko. Vrlo je malo stvari vrlo glatko. Pa ako pokušate s pitanjima: “Kolika je površina šenona?” Mjerit ćete i mjeriti i mjeriti. Kad god se približite, postaje veća, sve do vrlo, vrlo malih razmaka. Kolika je duljina obalne crte ovih jezera? Što bliže mjerite, to je dulja. Koncept duljine obalne crte, koji se čini tako prirodan, jer se navodi na mnogo mjesta, je zapravo zabluda; tako nešto ne postoji. Morate tome drugačije pristupiti.
What good is that, to know these things? Well, surprisingly enough, it's good in many ways. To begin with, artificial landscapes, which I invented sort of, are used in cinema all the time. We see mountains in the distance. They may be mountains, but they may be just formulae, just cranked on. Now it's very easy to do. It used to be very time-consuming, but now it's nothing. Now look at that. That's a real lung. Now a lung is something very strange. If you take this thing, you know very well it weighs very little. The volume of a lung is very small, but what about the area of the lung? Anatomists were arguing very much about that. Some say that a normal male's lung has an area of the inside of a basketball [court]. And the others say, no, five basketball [courts]. Enormous disagreements. Why so? Because, in fact, the area of the lung is something very ill-defined. The bronchi branch, branch, branch and they stop branching, not because of any matter of principle, but because of physical considerations: the mucus, which is in the lung. So what happens is that in a way you have a much bigger lung, but it branches and branches down to distances about the same for a whale, for a man and for a little rodent.
Što će vam to znanje? Pa, nećete vjerovati, korisno je na mnogo načina. Za početak, umjetni pejzaži, koje sam na neki način smislio, koriste se u kinematografiji sve vrijeme. Vidimo planine u daljini. To mogu biti planine, a mogu biti i samo pokrenute formule. Sad je to vrlo lako učiniti. Nekad je to bilo vrlo dugotrajno, sad to nije ništa. Pogledajte ovo. To su prava pluća. Pluća su nešto vrlo čudnovato. Uzmete li ih u ruke, znat ćete da vrlo malo teže. Volumen pluća je vrlo malen, no što je s površinom pluća? Anatomi su o tome puno raspravljali. Neki kažu da normalna muška pluća imaju površinu unutrašnjeg košarkaškog igrališta. A drugi kažu, ne, pet takvih igrališta. Poprilično neslaganje. A zašto? Zato što je površina pluća nešto što je vrlo loše definirano. Bronhiji se granaju, granaju, granaju i prestaju se granati, ne zbog nekog principa, već iz fizičkog razloga: zbog sluzi koja je u plućima. Pa se događa da možete imati i daleko veća pluća, no ona će se granati i granati do otprilike istih distanci i kod kita i kod čovjeka, kao i kod sitnog glodavca.
Now, what good is it to have that? Well, surprisingly enough, amazingly enough, the anatomists had a very poor idea of the structure of the lung until very recently. And I think that my mathematics, surprisingly enough, has been of great help to the surgeons studying lung illnesses and also kidney illnesses, all these branching systems, for which there was no geometry. So I found myself, in other words, constructing a geometry, a geometry of things which had no geometry. And a surprising aspect of it is that very often, the rules of this geometry are extremely short. You have formulas that long. And you crank it several times. Sometimes repeatedly: again, again, again, the same repetition. And at the end, you get things like that.
A kakve koristi od toga? Pa začudo, nećete vjerovati, anatomi su imali vrlo slabu predodžbu strukture pluća, sve donedavno. I mislim da je moja matematika, za divno čudo, bila od velike pomoći kirurzima koji su proučavali plućne bolesti kao i bolesti bubrega i svih sustava koji se granaju, a za koje nije bilo geometrije. Tako sam se takoreći našao u konstruiranju geometrije, geometrije stvari koje nisu imale geometriju. Tu je začuđujuće da su vrlo često pravila te geometrije izuzetno kratka. Imate ovoliko duge formule. Zavrtite ih nekoliko puta. Ponekad ponavljano: opet, opet i opet isto ponavljanje. Na kraju, dobijete nešto poput ovog. Ovaj oblak je u potpunosti, sto posto umjetan.
This cloud is completely, 100 percent artificial. Well, 99.9. And the only part which is natural is a number, the roughness of the cloud, which is taken from nature. Something so complicated like a cloud, so unstable, so varying, should have a simple rule behind it. Now this simple rule is not an explanation of clouds. The seer of clouds had to take account of it. I don't know how much advanced these pictures are. They're old. I was very much involved in it, but then turned my attention to other phenomena.
Dobro, 99,9. A jedini prirodni dio je broj, hrapavost tog oblaka, koji je uzet iz prirode. Nešto toliko komplicirano kao oblak, tako nestabilno, tako varirajuće, treba nositi u sebi jednostavno pravilo. A to jednostavno pravilo nije objašnjenje oblaka. Promatrač oblaka mora to uzeti u obzir. Ne znam koliko su napredne ove slike -- stare su. Prilično sam se bavio time, a onda pažnju posvetio drugim fenomenima.
Now, here is another thing which is rather interesting. One of the shattering events in the history of mathematics, which is not appreciated by many people, occurred about 130 years ago, 145 years ago. Mathematicians began to create shapes that didn't exist. Mathematicians got into self-praise to an extent which was absolutely amazing, that man can invent things that nature did not know. In particular, it could invent things like a curve which fills the plane. A curve's a curve, a plane's a plane, and the two won't mix. Well, they do mix. A man named Peano did define such curves, and it became an object of extraordinary interest. It was very important, but mostly interesting because a kind of break, a separation between the mathematics coming from reality, on the one hand, and new mathematics coming from pure man's mind. Well, I was very sorry to point out that the pure man's mind has, in fact, seen at long last what had been seen for a long time. And so here I introduce something, the set of rivers of a plane-filling curve. And well, it's a story unto itself. So it was in 1875 to 1925, an extraordinary period in which mathematics prepared itself to break out from the world. And the objects which were used as examples, when I was a child and a student, as examples of the break between mathematics and visible reality -- those objects, I turned them completely around. I used them for describing some of the aspects of the complexity of nature.
A ima još jedna prilično interesantna stvar. Jedan od dramatičnih događaja u povijesti matematike, kojeg mnogo ljudi ne razumije, dogodio se prije oko 130, 145 godina. Matematičari su počeli kreirati oblike koji nisu postojali. Matematičari su bili puni samohvale do mjere koja je bila zapanjujuća, da čovjek može smisliti stvari koje priroda nije poznavala. Točnije, mogao je smisliti nešto poput krivulje koja ispunjava ravninu. Krivulja je krivulja, a ravnina ravnina, i to dvoje se ne želi miješati. A ipak se miješaju. Čovjek po imenu Peano definirao je takve krivulje, što je postalo predmetom nevjerojatnog zanimanja. Bilo je to vrlo važno, no uglavnom zanimljivo zbog svojevrsnog raskola, razdvajanja matematike koja dolazi iz stvarnosti, i nove matematike koja dolazi iz čistog ljudskog uma. Nažalost sam pokazao da je čisti ljudski um ustvari konačno uvidio ono što je već odavno bilo viđeno. Tako ovdje uvodim nešto, niz rijeka krivulje koja ispunjava ravninu. No dobro, to je priča za sebe. Bilo je to od 1875. do 1925., u fascinantnom razdoblju u kojem se matematika pripremala za odvajanje od svijeta. A predmeti korišteni dok sam bio đak i student kao primjeri raskola matematike i vidljive realnosti -- te predmete, potpuno sam ih obrnuo. Koristio sam ih za opisivanje nekih aspekata kompleksnosti prirode.
Well, a man named Hausdorff in 1919 introduced a number which was just a mathematical joke, and I found that this number was a good measurement of roughness. When I first told it to my friends in mathematics they said, "Don't be silly. It's just something [silly]." Well actually, I was not silly. The great painter Hokusai knew it very well. The things on the ground are algae. He did not know the mathematics; it didn't yet exist. And he was Japanese who had no contact with the West. But painting for a long time had a fractal side. I could speak of that for a long time. The Eiffel Tower has a fractal aspect. I read the book that Mr. Eiffel wrote about his tower, and indeed it was astonishing how much he understood.
1919., čovjek imenom Hausdorff uveo je broj koji je smatran samo matematičkom šalom, a ja sam otkrio da je taj broj dobra mjera za hrapavost. Kad su to moji matematičari čuli, rekli su, “Ne budi blesav, to je samo...” Pa zapravo, nisam bio blesav. Slavni slikar Hokusai to je dobro znao. Ono na tlu su alge. On nije poznavao tu matematiku; toga još nije bilo. A on je bio Japanac koji nije imao kontakta sa Zapadom. No slikarstvo je odavna imalo fraktalnu stranu. O tome bih mogao dugo pričati. Eiffelov toranj ima fraktalni aspekt. Pročitao sam knjigu g. Eiffela o njegovom tornju, i zaista je fascinantno koliko je toga shvaćao.
This is a mess, mess, mess, Brownian loop. One day I decided -- halfway through my career, I was held by so many things in my work -- I decided to test myself. Could I just look at something which everybody had been looking at for a long time and find something dramatically new? Well, so I looked at these things called Brownian motion -- just goes around. I played with it for a while, and I made it return to the origin. Then I was telling my assistant, "I don't see anything. Can you paint it?" So he painted it, which means he put inside everything. He said: "Well, this thing came out ..." And I said, "Stop! Stop! Stop! I see; it's an island." And amazing. So Brownian motion, which happens to have a roughness number of two, goes around. I measured it, 1.33. Again, again, again. Long measurements, big Brownian motions, 1.33. Mathematical problem: how to prove it? It took my friends 20 years. Three of them were having incomplete proofs. They got together, and together they had the proof. So they got the big [Fields] medal in mathematics, one of the three medals that people have received for proving things which I've seen without being able to prove them.
To je zbrka, Brownova petlja. [Brownov klaster] Jednog dana sam odlučio -- negdje na polovici svoje karijere. Bio sam ograničen s puno stvari u poslu -- odlučio sam se iskušati. Bi li mogao samo gledati nešto što su već svi dugo vremena promatrali i pronaći nešto geometrijski novo? Tako sam promatrao ove stvari nazvane Brownovo gibanje -- samo ide uokolo. Neko sam se vrijeme igrao njime, i natjerao ga da se vrati na početak. Onda bih rekao svom asistentu, “Ne vidim ništa; možeš li to nacrtati?” I on je to nacrtao, sve je stavio unutra. Rekao je: “Pa ispalo je ovo...” Povikao sam, “Stani, stani, stani! Vidim, to je otok.” Upravo nevjerojatno. Dakle, Brownovo gibanje, koje ima hrapavost broja dva, ide uokolo. Izmjerio sam ga, 1,33. Opet, i opet, i opet. Duga mjerenja, velika Brownova gibanja, 1,33. Matematički problem: kako to dokazati? Mojim je prijateljima trebalo 20 godina. Trojica su imala nepotpune dokaze. Sastali su se i zajedno našli dokaz. Pa su dobili veliku [Fieldsovu] medalju u matematici, jednu od tri medalje koju su drugi dobili jer su dokazali stvari koje sam vidio ali ih nisam mogao dokazati.
Now everybody asks me at one point or another, "How did it all start? What got you in that strange business?" What got you to be, at the same time, a mechanical engineer, a geographer and a mathematician and so on, a physicist? Well actually I started, oddly enough, studying stock market prices. And so here I had this theory, and I wrote books about it -- financial prices increments. To the left you see data over a long period. To the right, on top, you see a theory which is very, very fashionable. It was very easy, and you can write many books very fast about it. (Laughter) There are thousands of books on that. Now compare that with real price increments. Where are real price increments? Well, these other lines include some real price increments and some forgery which I did. So the idea there was that one must be able to -- how do you say? -- model price variation. And it went really well 50 years ago. For 50 years, people were sort of pooh-poohing me because they could do it much, much easier. But I tell you, at this point, people listened to me. (Laughter) These two curves are averages: Standard & Poor, the blue one; and the red one is Standard & Poor's from which the five biggest discontinuities are taken out. Now discontinuities are a nuisance, so in many studies of prices, one puts them aside. "Well, acts of God. And you have the little nonsense which is left. Acts of God." In this picture, five acts of God are as important as everything else. In other words, it is not acts of God that we should put aside. That is the meat, the problem. If you master these, you master price, and if you don't master these, you can master the little noise as well as you can, but it's not important. Well, here are the curves for it.
Svi me prije ili kasnije pitaju, “Kako je to sve započelo? Što vas je dovelo u te neobične vode? Što sve morate istovremeno biti - strojarski inženjer, geograf, matematičar i fizičar?” Zapravo sam počeo, nećete vjerovati, proučavajući cijene na burzi. I tako sam tu razvio ovu teoriju i o njoj pisao knjige -- rast financijskih cijena. Lijevo su podaci prikupljani dugo vremena. Na vrhu zdesna, vidite teoriju koja je vrlo, vrlo moderna. Jednostavna je i možete o njoj vrlo brzo napisati puno knjiga. (Smijeh) Na tisuće je knjiga napisano o tome. Sad to usporedite sa stvarnim pomacima cijena. Gdje su stvarni pomaci cijena? Dakle, ove druge linije uključuju neke stvarne pomake cijena i neke lažne koje sam ja kreirao. Tako je tu ideja bila da se mora moći -- kako bi to rekli? -- dati model varijacije cijena. I to je stvarno dobro išlo pred 50 godina. Čitavih 50 godina su me pomalo omalovažavali jer se to moglo puno lakše izvesti. Ali kažem vam, tada su me ljudi slušali. (Smijeh) Ove dvije krivulje su prosječne vrijednosti: [indeks agencije] S&P, plava, a crvena [indeks agencije] S&P bez pet najvećih diskontinuiteta. Diskontinuiteti su gnjavaža pa se u mnogim analizama cijena oni izostavljaju. “Dakle, viša sila.” A ono što ostaje je malo besmisleno. Na ovoj je slici pet viših sila važno koliko i sve ostalo. Drugim riječima, višu silu ne bi trebali držati po strani. To je srž, to je problem. Ovladate li time, ovladat ćete cijenom, a ne ovladate li time, možete se do mile volje uspješno baviti šumom, ali to nije važno. Dakle, tu su krivulje za to.
Now, I get to the final thing, which is the set of which my name is attached. In a way, it's the story of my life. My adolescence was spent during the German occupation of France. Since I thought that I might vanish within a day or a week, I had very big dreams. And after the war, I saw an uncle again. My uncle was a very prominent mathematician, and he told me, "Look, there's a problem which I could not solve 25 years ago, and which nobody can solve. This is a construction of a man named [Gaston] Julia and [Pierre] Fatou. If you could find something new, anything, you will get your career made." Very simple. So I looked, and like the thousands of people that had tried before, I found nothing.
Sad stižem do konačne stvari, a to je set koji nosi moje ime. Na neki način, moja životna priča. Bio sam adolescent za njemačke okupacije Francuske. Budući da sam mislio da bih mogao nestati za koji dan ili tjedan, imao sam vrlo velike snove. A nakon rata, ponovno sam vidio ujaka. Moj mi je ujak, čuveni matematičar, rekao, “Gle, postoji problem koji nisam uspio riješiti prije 25 godina i nitko ga ne može riješiti. To je konstrukcija matematičara [Gaston] Julia i [Pierre] Fatou. Kad bi mogao otkriti nešto novo, bilo što, napravio bi si karijeru.” Vrlo jednostavno. Proučavao sam to, i poput tisuća njih prije mene, nisam našao ništa.
But then the computer came, and I decided to apply the computer, not to new problems in mathematics -- like this wiggle wiggle, that's a new problem -- but to old problems. And I went from what's called real numbers, which are points on a line, to imaginary, complex numbers, which are points on a plane, which is what one should do there, and this shape came out. This shape is of an extraordinary complication. The equation is hidden there, z goes into z squared, plus c. It's so simple, so dry. It's so uninteresting. Now you turn the crank once, twice: twice, marvels come out. I mean this comes out. I don't want to explain these things. This comes out. This comes out. Shapes which are of such complication, such harmony and such beauty. This comes out repeatedly, again, again, again. And that was one of my major discoveries, to find that these islands were the same as the whole big thing, more or less. And then you get these extraordinary baroque decorations all over the place. All that from this little formula, which has whatever, five symbols in it. And then this one. The color was added for two reasons. First of all, because these shapes are so complicated that one couldn't make any sense of the numbers. And if you plot them, you must choose some system. And so my principle has been to always present the shapes with different colorings because some colorings emphasize that, and others it is that or that. It's so complicated.
A onda se pojavilo računalo pa sam odlučio primijeniti računalo, ne za nove probleme u matematici -- kao ovo miganje, to je novi problem -- već za stare probleme. I pošao sam od onog što se naziva realni brojevi, što su točke na pravcu, prema imaginarnim, kompleksnim brojevima, što su točke u ravnini, a to je ono što bi se trebalo raditi, i pojavio se ovaj oblik. Ovaj je oblik izuzetno složen. Jednadžba koja se tu skriva je z prelazi u z na kvadrat, plus c. Tako jednostavno, tako suhoparno. Tako nezanimljivo. Sad je izvrtite jednom, dvaput: dvaput -- čudesa nastaju. Mislim, nastaje ovo. Ne želim objašnjavati te stvari. Nastaje ovo, i ovo. Oblici tako zamršeni, a tako skladni i lijepi. Ovo nastaje ponavljano, opet, opet i opet. A to je bilo jedno od mojih velikih otkrića da su ti otoci bili isti kao i čitava cjelina, manje-više. A onda posvuda dobijete ove neobične barokne uzorke. I sve to iz ove male formule, koja ima nekoliko, pet simbola, u sebi. A onda ovo. Boja je dodana iz dva razloga. Prvo, jer su ti oblici toliko složeni da se ne bi mogao uhvatiti smisao tih brojeva. A ako ih crtate ploterom, morate izabrati neki sistem. Tako je moj princip bio uvijek prikazati oblike u različitim bojama jer neke boje naglašavaju ovo, a druge ono. To je tako zamršeno.
(Laughter)
(Smijeh)
In 1990, I was in Cambridge, U.K. to receive a prize from the university, and three days later, a pilot was flying over the landscape and found this thing. So where did this come from? Obviously, from extraterrestrials. (Laughter) Well, so the newspaper in Cambridge published an article about that "discovery" and received the next day 5,000 letters from people saying, "But that's simply a Mandelbrot set very big."
1990., bio sam u Cambridgeu, U.K. i primao nagradu tog sveučilišta, a tri dana kasnije, pilot je nadlijetao područje i našao ovo. Odakle je to došlo? Očito, od izvanzemaljaca. (Smijeh) Dakle, novine u Cambridgeu objavile su članak o tom “otkriću” i sljedećeg dana primile 5000 pisama u kojima je pisalo “Ma to je samo vrlo velik Mandelbrotov set.”
Well, let me finish. This shape here just came out of an exercise in pure mathematics. Bottomless wonders spring from simple rules, which are repeated without end.
Dakle, da završim. Ovaj je oblik proizašao iz vježbanja čiste matematike. Neizmjerna čuda proizlaze iz jednostavnih pravila, što se ponavljaju bez kraja.
Thank you very much.
Najljepša vam hvala.
(Applause)
(Pljesak)