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.
Mnogo vam hvala. Oprostite što sedim, ja sam jako star. (Smeh) Znači, tema o kojoj ću razgovarati je u izvesnom smislu dosta neobična jer je vrlo stara. Hrapavost je deo ljudskog života, i tako će biti za sva vremena. Još su drevni pisci pisali o tome. Nju je uglavnom nemoguće kontrolisati. I u izvesnom smislu, čini se da je krajnje složena, samo zbrka i nered. Postoji mnogo različitih vidova nereda. Zapravo, zahvaljujući potpuno srećnom slučaju, pre nekoliko godina sam se uključio u izučavanje ovog vida kompleksnosti. Na moje kranje čuđenje, pronašao sam tragove – veoma snažne tragove, moram reći – reda u toj hrapavosti. Stoga ću danas da vam predstavim nekoliko primera onoga što to predstavlja. Više mi se sviđa reč hrapavost od nepravilnosti, jer nepravilnost – za nekoga ko je učio latinski u dalekoj mladosti – znači nešto što je suprotno pravilnosti. Ali nije. Pravilnost je suprotnost hrapavosti jer je osnovni aspekt sveta veoma 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)
Dopustite mi da vam pokažem nekoliko predmeta. Neki od njih su veštački. Neki su stvarni, u izvesnom smislu. Recimo, ovaj je pravi. Ovo je karfiol. Ali zašto pokazujem karfiol, jedno jako obično i drevno povrće? Zbog toga što, koliko god da je staro i drevno, ono je veoma komplikovano i veoma jednostavno u isto vreme. Ako pokušate da mu izmerite težinu, naravno, to je jako jednostavno. A kada ga jedete, težina je bitna. No, pretpostavite da pokušate da izmerite njegovu površinu. Da, to je jako zanimljivo. Ako oštrim nožem odsečete jedan od njegovih cvetova i pogledate ga odvojeno, pomislićete na celi karfiol, samo manji. I onda nastavite da režete, dalje, dalje, dalje, dalje, dalje, dalje, dalje, dalje. I stalno dobijate male karfiole. Iskustvo čovečanstva pokazuje da oduvek postoje oblici koji poseduju ovakav poseban kvalitet, gde svaki deo izgleda kao celina, samo manja. A šta je čovečanstvo učinilo sa time? Veoma, veoma malo. (Smeh)
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.
Ono što sam ja zapravo učinio je da sam proučavao ovaj problem i otkrio nešto sasvim iznenađujuće. Otkrio sam da je moguće izmeriti hrapavost nekim brojem, brojem, 2.3, 1.2, a ponekad i nekim većim. Jedan dan, moj prijatelj je, kako bi me zavitlavao, doneo sliku i rekao „Kolika je hrapavost ove obline?“ Odgovorio sam, „Pa, nešto manja od 1.5.“ Bila je 1.48. Znači, uopšte nije bilo potrebno mnogo vremena za to, jer sam toliko dugo posmatrao te stvari. Znači, ovi brojevi su brojevi koji označavaju hrapavost ovih površina. Želim da što pre kažem da su ove površine potpuno veštačke. Napravljene su kompjuterski. I jedini unos je broj. A taj broj je hrapavost. I tako sam na levoj strani uzeo hrapavost preuzetu iz mnogih pejzaža. Na desnoj strani, uzeo sam malo više hrapavosti. I tako oko, nakon nekog vremena, može jako dobro napraviti razliku između jednoga i drugoga.
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.
Čovečanstvo mora da nauči o merenju hrapavosti. Ovo je hrapavo, ovo je donekle glatko, a ovo je savršeno glatko Izuzetno malo toga je izuzetno glatko. Pokušajte onda da upitate: kolika je površina karfiola? Možete meriti i meriti i meriti. Svaki put kada se približite on postane još veći, čak i do najmanjih razmaka. Kolika je dužina obale ovih jezera? Što bliže merite, biće sve duža. Koncept dužine obale, koja se čini tako prirodnom jer se u mnogim slučajevima uzima zdravo za gotovo, zapravo je potpuna obmana; ne postoji ništa takvo. Morate da pokušate na drugačiji način.
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.
Kakva je korist znati sve ove stvari? Hm, dovoljno iznenađujuće, korisno je na mnogo načina. Za početak, veštački pejzaži koje sam na neki način izmislio, koriste se u kinematografiji sve vreme. Vidimo planine u daljini. Mogle bi biti planine, ali bi mogle biti i formule, samo sklepane. U današnje vreme je to jako jednostavno učiniti. Nekada je za to bilo potrebno mnogo vremena, ali sada je to sitnica. Pogledajte. Ovo su prava pluća. Pluća su nešto jako čudno. Ako ih uzmete, znate jako dobro da je njihova težina mala. Zapremina pluća je veoma mala. Ali kolika je površina pluća? Anatomi su se mnogo svađali oko toga. Neki kažu da pluća normalnog muškarca imaju površinu unutrašnjosti košarkaškog igrališta. A oni drugi kažu – ne, pet košarkaških igrališta. Ogromna razlika u mišljenju. Zbog čega? Zato što je, zapravo, površina pluća nešto što je jako loše definisano. Bronhije se razgranavaju, razgranavaju, razgranavaju. I onda prestanu da se granaju, ne zbog bilo kog posebnog principa, već zbog fizičkih okolnosti, sluzi koja se nalazi u plućima. Znači, ono što se dešava jeste da na taj način imate mnogo veća pluća, ali ako se razgranavaju i razgranavaju, sve do rastojanja koja su gotovo jednaka kod kita, čoveka ili malog glodara.
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.
Ali, kakva je korist od toga? Pa, dovoljno iznenađujuće, dovoljno začuđujuće, je da su anatomi imali jako bledu ideju o strukturi pluća sve donedavno. A ja mislim da je moja matematika, sasvim začuđujuće, bila od velike pomoći hirurzima koji izučavaju bolesti pluća kao i bolesti bubrega, sve te sisteme koji se razgranjavaju, za koje nije postojala nikakva geometrija. I tako sam se našao kako, drugim rečima, konstruišem geometriju, geometriju stvari koje nemaju geometriju. A iznenađujući aspekt toga je da su veoma često pravila ove geometrije izuzetno kratka. Imate formule toliko dugačke. I onda ih izvršite nekoliko puta. Ponekad to ponovite više puta. Isto ponavljanje. I na kraju dobijete nešto poput toga.
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.
Ovaj oblak je potpuno, stoprocentno veštački. Pa dobro, 99.9. A jedini deo koji je prirodan je broj, hrapavost oblaka, koji je preuzet iz prirode. Iza nečega tako komplikovanog poput oblaka, tako nestalnog, tako promenljivog, bi trebalo postojati jednostavno pravilo. Ali to jednostavno pravilo nije objašnjenje za oblake. Posmatrač oblaka mora to uzeti u obzir. Nisam siguran koliko su unapređene ove slike, jer su stare. Mnogo sam se bavio njima, . ali onda sam preusmerio pažnju ne druge fenomene.
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.
Ovde imamo još jednu stvar koja je dosta zanimljiva. Jedan od dramatičnih događaja u istoriji matematike, a koji veliki broj ljudi uopšte ne ceni, desio se pre nekih 130, 145 godina. Matematičari su počeli da stvaraju oblike koji nisu postojali. Matematičari su počeli sa hvalospevima samim sebi do te mere gde je potpuno neverovatno da čovek može izumeti stvari koje sama priroda nije poznavala. A posebno da može izumeti stvari poput krive koja ispunjava ravan Kriva je kriva, ravan je ravan, i jedno sa drugim se ne meša. Ali, ipak se mešaju. Čovek po imenu Peano je definisao takve krive, koje su postale predmet neverovatnog interesovanja. To je bilo jako bitno, ali uglavnom zanimljivo zbog izvesnog prekida, razdvajanja između matematike koja proizilazi iz realnosti, s jedne strane i nove matematike koja dolazi iz čistog uma čoveka. Bilo mi je jako žao što sam morao ukazati da je čisti um čoveka zapravo konačno video ono što se moglo videti već dugo vremena. I tako sam ovde uveo nešto novo, niz reka krive koja ispunjava ravan. Ali, to je priča sama za sebe. Dakle, između 1875. i 1925., u to neverovatno doba matematika se pripremala da se odvoji od sveta. A predmete koji su korišćeni kao primeri, u vreme kada sam ja bio dete i student, primeri prekida između matematike i vidljive realnosti – te predmete sam potpuno preokrenuo. Koristio sam ih kako bih opisao neke od aspekata složenosti 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.
Čovek po imenu Hausdorf je 1919. godine uveo broj koji je bio samo matematička šala. A ja sam otkrio da je taj broj dobra mera hrapavosti. Kada sam ga prvi put saopštio mojim prijateljima matematičarima rekli su, "Ne budi lud. To je tek nešto (blesavo)". No, zapravo, uopšte nije bilo blesavo. Čuveni slikar Hokusai je to jako dobro znao. Ovo su alge. On nije bio upoznat sa matematikom; u to vreme još nije postojala. A on je bio Japanac bez ikakvog kontakta sa zapadnim svetom. No, njegove su slike odavno imale odliku fraktala. Mogao bih nadugo govoriti o ovome. Ajfelov toranj ima fraktalni aspekt. A pročitao sam i knjigu koju je g-din Ajfel napisao o svom tornju. I, zaista, bilo je zapanjujuće koliko mnogo je on razumeo.
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, zbrka, zbrka, Braunovo kretanje. Jednog dana, na pola svoja karijere, toliko toga me je usporavalo u mom poslu, i odlučio sam da se podvrgnem testu. Da li bih mogao gledati u nešto što svi posmatraju već jako dugo vremena i pronaći nešto dramatično novo? I tako sam gledao u ove stvari koje se nazivaju Braunovim kretanjem – i tako u krug. Neko vreme sam se igrao sa tim i vratio je na početak. Govorio sam svome asistentu „Ništa ne vidim. Možeš li ti to naslikati?“ I on je naslikao, što znači da je sve stavio unutra. Rekao je: „Hm, ovo je izašlo....“ A ja sam rekao „Stani! Stani! Stani! Vidim ga, to je ostrvo." Zapanjujuće. Znači, Braunovo kretanje, koja ima hrapavost broja dva, se okreće. Izmerio sam ga, 1.33. Opet, i opet, i opet. Duga merenja, velika Braunova kretanja, 1.33. Matematički problem: kako ga dokazati? Mojim prijateljima je trebalo 20 godina. Trojica njih su imali nepotpune dokaze. Oni su se sastali i tek zajedno su imali dokaz. Tako su dobili veliku medalju u matematici, jednu od tri priznanja koja pojedinci primaju za dokazivanje stvari koje sam video ali nisam bio u stanju da dokažem.
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.
I sada me svi pre ili kasnije pitaju „Kako je to sve počelo? Šta te je navelo na taj čudni posao?“ Šta me je navelo da istovremeno budem mehanički inženjer, geograf i matematičar, i tako dalje, fizičar? Hm, zapravo sam počeo izučavati cene na berzi. I tako sam imao ovu teoriju, i pisao sam knjige o tome. Rast finansijskih cena. Na levoj strani vidite podatke prikupljane tokom dugo vremena. Na desnoj, na vrhu, vidite teoriju koja je veoma, veoma u modi. Bilo je jako jednostavno i jako brzo možete napisati mnogo knjiga o tome. (Smeh) Postoji na hiljade knjiga o tome. Sada uporedite to sa stvarnim uvećanjem cena i gde su prava uvećanja cena? Ove druge linije uključuju neka realna uvećanja cena, i neka lažna, koja sam ja uneo. Znači, ideja je bila da je moguće – kako biste to rekli? – napraviti model varijacije cena. I to je išlo jako dobro pre 50 godina. Pedeset godina ljudi su me na neki način omalovažavali jer su oni to mogli uraditi na mnogo, mnogo jednostavniji način. Ali da vam kažem, ovoga puta, ljudi su me slušali. (Smeh) Ove dve krive su prosečne vrednosti. S&P (agencija za kreditne rejtinge), plave boje. A crvena je S&P, iz koje je pet najvećih diskontinuiteta izdvojeno. No, diskontinuiteti su smetnja. U mnogim studijama cena, oni se ostavljaju po strani. "Pa, više sile. I imate malo besmislenost koje je ostalo. Više sile." Na ovoj slici pet takvih sila su podjednako važne kao i sve ostalo. Drugim rečima, nije viša sila to što bismo trebali staviti po strani. To je suština, problem. Ako ovladate njima, ovladali ste cenom. Ako ne ovladate njima, možete ovladati nad ovo malo šuma koliko god želite, no to nije važno. Evo, ovde su krive koje to pokazuju.
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.
Sada ću preći na poslednju stvar, a to je niz od kog je sačinjeno moje ime. Na neki način, to je priča mog života. Svoje doba adolescencije sam proveo tokom nemačke okupacije Francuske. I s obzirom da sam mislio kako bih mogao nestati kroz dan ili nedelju, imao sam velike snove. I nakon rata sam ponovo video svog strica. Moj stric je bio jako istaknut matematičar i rekao mi je „Pazi, ima jedan problem koji nisam mogao rešiti pre 25 godina, i koji niko ne može da reši. To je konstrukcija čoveka po imenu (Gaston) Julia i (Pierre) Fatou. Ako bi uspeo da nađeš nešto novo, bilo šta, napravićeš karijeru“. Veoma jednostavno. I tako sam počeo tražiti, i kao i hiljade onih koji su pokušali pre mene, nisam ništa našao.
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 pojavio kompjuter. I ja sam odlučio da primenim kompjuter, ne na nove matematičke probleme – poput ovog mrdanja ovde, to je novi problem – već na stare probleme. I krenuo sam od onoga što se zove realni brojevi, koji predstavljaju tačke na liniji, ka imaginarnim, složenim brojevima, koji su tačke na ravni, i to je ono što treba da se tu uradi. I dobio sam ovaj oblik. Ovaj oblik je neverovatno komplikovan. Ovde je skrivena jednačina, z ide u z na kvadrat, plus c. Jako je jednostavno, jako suvo. I tako nezanimljivo. Okrenite ovaj kotur jednom, drugi put, još jednom, i izaći će čudo. Mislim, ovo će izaći. Ne želim da objašnjavam ove stvari. Pojavljuje se ovo. I ovo. Oblici koji su tako komplikovani, takvog sklada i lepote. Ovo izlazi ponovo, iznova i iznova. I to je bilo jedno od mojih glavnih otkrića da su ova ostrva ista kao i cela velika stvar, manje ili više. A onda imate ove neobične barokne dekoracije posvuda. I sve to iz ove male formule, koja se sastoji od koliko god, nekih pet simbola. A potom ova. Boja je dodana iz dva razloga. Najpre zbog toga što su ovi oblici previše komplikovani da bi iko mogao videti bilo kakav smisao u brojevima. A ako ih unosite u grafikon, morate odabrati neki sistem. A moj princip je oduvek bio da predstavim oblike različitim bojama, zato što neke boje naglašavaju ovo, a neke druge ovo ili ono. Tako je komplikovano.
(Laughter)
(Smeh)
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. godine sam bio u Kembridžu, u Ujedinjenom Kraljevstvu kako bih primio nagradu univerziteta. I tri dana nakon toga, jedan pilot je leteo iznad pejzaža i otkrio ovo. Odakle se ovo pojavilo? Očigledno, od vanzemaljaca. (Smeh) Tako su novine u Kembridžu objavile članak o „otkriću“ i sledeći dan primile 5.000 pisama od čitalaca koji su rekli „Ali to je naprosto veoma veliki 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.
Dopustite mi da završim. Ovaj oblik ovde je nastao kao vežba u čistoj matematici. Beskrajna čuda izviru iz jednostavnih pravila, koja se beskrajno ponavljaju.
Thank you very much.
Hvala vam mnogo.
(Applause)
Aplauz.