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.
Ďakujem veľmi pekne. Prosím ospravedlňte, že sedím: Som veľmi starý. (Smiech) Nuž, téma o ktorej budem rozprávať, je v istom zmysle veľmi špecifická, pretože je veľmi stará. Drsnosť je súčasť ľudského života od nepamäti. Už dávni autori o nej písali. Je veľmi nekontrolovateľná. A v istom zmysle sa zdalo, že je to extrém zložitosti, proste neporiadok, neporiadok, neporiadok. Sú rôzne druhy neporiadku. Takže, úplnou náhodou som sa dostal veľmi dávno k štúdiu tejto formy zložitosti. A na moje hlboké počudovanie, som našiel stopy -- veľmi silné stopy, musím povedať -- poriadku v tej drsnosti. A dnes by som vám rád prezentoval zopár príkladov toho, čo to predstavuje. Preferujem slovo drsnosť pred slovom nepravidelnosť, pretože nepravidelnosť -- pre niekoho, kto mal latinčinu, v mojej dávnej mladosti -- znamená opak pravidelnosti. Ale tak to nie je. Pravidelnosť je opakom drsnosti, pretože základný aspekt sveta je veľmi drsný.
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)
Ukážem vám pár objektov. Niektoré z nich sú umelé. Niektoré sú veľmi reálne, v istom zmysle. Toto je ten reálny. Je to karfiol. Ale prečo ukazujem karfiol, veľmi obyčajnú a pradávnu zeleninu? Pretože tak aký je starý a pradávny, je aj veľmi komplikovaný a veľmi jednoduchý zároveň. Ak ho skúsite odvážiť, samozrejme, je ľahké ho odvážiť. A keď ho jete, záleží na váhe. Ale čo keď skúsite zmerať jeho povrch. Nuž, to je veľmi zaujímavé. Ak odrežete ostrým nožom jeden kvietok karfiolu a pozriete sa naň samostatne, vidíte celý karfiol, len menší. A tak ho rozkrojíte znova, znova, znova, znova, znova, znova, znova, znova, znova. A stále dostávate malé karfioly. Zo skúsenosti ľudstva vieme, že sú isté tvary, ktoré majú túto zvláštnu vlastnosť, že každá časť je ako celok, len menšia. A čo s tým ľudstvo spravilo? Veľmi, veľmi málo. (Smiech)
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.
Tak som sa do toho pustil, študoval som tento problém a objavil som niečo dosť prekvapujúce. Že sa dá merať drsnosť číslom, číslom, 2,3, 1,2 a niekedy oveľa viac. Jedného dňa, jeden priateľ, aby ma podpichol, priniesol obrázok a pýta sa, "Aká je drsnosť tejto krivky?" Povedal som, "Nuž, tesne pod 1,5" Bolo to 1.48. Nezabralo to žiadny čas. Už som sa na tieto veci pozeral tak dlho. Tieto čísla sú čísla, ktoré označujú drsnosť týchto povrchov. Rýchlo dodám, že tieto povrchy sú úplne umelé. Vytvoril ich počítač. A jediným vstupom je číslo. A to číslo je drsnosť. Naľavo, som vzal drsnosť prebranú z prírody. Vpravo je vyššia drsnosť. A oko, po nejakej dobe, vie tieto dva pomerne dobre rozlíšiť.
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.
Ľudstvo sa muselo naučiť merať drsnosť. Toto je veľmi drsné, toto je pomerne hladké a toho je úplne hladké. Veľmi málo vecí je hladkých. A potom, ak sa opýtate otázky: aký je povrch karfiolu? Nuž, meriate, meriate a meriate. Vždy, keď sa priblížite, povrch sa zväčšuje, až k veľmi, veľmi malým vzdialenostiam. Aká je dĺžka pobrežia týchto jazier? Čím z bližšej vzdialenosti sa pozeráte, tým je dlhšia. Koncept dĺžky pobrežia, čo sa zdá také prirodzené, pretože sa udáva na veľa miestach, je, vlastne, úplná lož; nič také nie je. Musíte to robiť inak.
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.
Na čo je to teda dobré, vedieť tieto veci? Nuž, prekvapujúco, je to užitočné mnohými spôsobmi. Na začiatok, umelé krajiny, ktoré som tak trochu vymyslel, sa používajú v kine vlastne stále. Vidíme hory na horizonte. Môžu to byť hory, ale môžu to byť len vzorce, len uletené. Je to vlastne veľmi jednoduché. Bývalo to veľmi časovo náročné, ale dnes je to nič. Pozrite sa na to. To sú skutočné pľúca. Pľúca sú niečo veľmi čudné. Ak vezmete túto vec, veľmi dobre viete, koľko váži. Objem pľúc je veľmi malý. Ale ako je to s povrchom pľúc? Anatómovia sa o tom dosť sporili. Niektorí vravia, že normálne mužské pľúca majú povrch basketbalového ihriska. A iní vravia, nie, piatich ihrísk. Úžasné nezrovnalosti. Prečo to? Lebo, v skutočnosti, povrch pľúc, je niečo veľmi zle definované. Priedušky sa vetvia, vetvia, vetvia. A potom sa prestanú vetviť, nie kvôli nejakému princípu, ale kvôli fyzikálnym okolnostiam, kvôli hlienu, ktorý je v pľúcach. Takýmto spôsobom máte oveľa väčšie pľúca, ktoré sa vetvia a vetvia, až k vzdialenostiam, ktoré sú zhruba rovnaké pre veľryby, pre ľudí a pre malé hlodavce.
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.
Na čo je to všetko dobré? Nuž, prekvapujúco, až udivujúco, anatómovia mali až donedávna veľmi chabé znalosti o štruktúre pľúc. A myslím, že moja matematika, prekvapujúco, veľmi pomohla chirurgom, ktorí študujú choroby pľúc a tiež ochorenia obličiek, všetkých týchto vetviacich sa systémov, pre ktoré nebola žiadna geometria. Inými slovami, ocitol som sa v konštruovaní geometrie, geometrie vecí, ktoré nemali geometriu. Prekvapujúcim aspektom toho je, že veľmi často, pravidlá tejto geometrie sú extrémne krátke. Máte takéto krátke vzorce. Párkrát ich pretočíte. Niekedy opakovane znova, znova, znova. To isté opakovanie. A na koniec dostanete veci ako toto.
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.
Tento oblak je úplne, 100% umelý. Nuž, 99,9. Jediná prirodzená časť je číslo, drsnosť oblaku, ktoré je prevzaté z prírody. Niečo tak komplikované ako oblak, tak nestále, tak premenlivé, by malo mať v pozadí jednoduché pravidlo. Toto jednoduché pravidlo nie je vysvetlením oblakov. Pozorovateľ oblakov to musel vziať do úvahy. Neviem nakoľko pokročilé sú tieto obrázky, sú staré. Bol som súčasťou tejto oblasti, ale potom som obrátil pozornosť na iné fenomény.
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.
Je ďalšia vec, ktorá je pomerne zaujímavá. Jedna z šokujúcich udalostí v histórii matematiky, nedoceňovaná mnohými ľuďmi, sa vyskytla asi pred 130 rokmi, pred 145 rokmi. Matematici začali tvoriť tvary, ktoré neexistovali. Matematici začali chváliť sami seba až do úplne neuveriteľných rozmerov, za to, že človek dokáže vymyslieť veci, ktoré príroda nepoznala. Napríklad, vytvorili veci ako krivku, ktorá vyplní rovinu. Krivka je krivka, rovina je rovina a to nejde dokopy. Nuž, ide. Muž, Peano definoval také krivky a stalo sa to objektom nevídaného záujmu. Bolo to veľmi dôležité, ale hlavne zaujímavé, lebo to bol istý druh zlomu, separácia medzi matematikou vychádzajúcej z reality na jednej strane a novou matematikou, vychádzajúcej z čistej ľudskej mysle. Nuž, veľmi nerád som vtedy pripomenul, že čistá ľudská myseľ vlastne, konečne uvidela, čo bolo k videniu už dlhý čas. Takže teraz niečo predstavím, súbor riek krivky vypĺňajúcej rovinu. A vlastne, je to príbeh sám o sebe. Takže, bolo to v rokoch 1875 až 1925, výnimočné obdobie, v ktorej sa matematici pripravovali uniknúť z tohto sveta. A objekty, ktoré používali ako príklady -- keď som bol dieťa a študent -- rozchodu medzi matematikou a viditeľnou realitou -- tieto objekty, som úplne otočil. Použil som ich na popísanie niektorých aspektov komplexity prírody.
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.
Nuž, istý Hausdorff v roku 1919 predstavil číslo, ktoré bolo len matematickým vtipom. A prišiel som na to, že toto číslo je dobrým meradlom drsnosti. Keď som to prvýkrát povedal priateľom matematikom, vraveli, "Nebuď blázon. To je len nejaká blbosť." Nuž, nakoniec, nebola to blbosť. Veľký maliar Hokusai to dobre poznal. Tie veci na zemi sú riasy. Nepoznal tú matematiku, ešte neexistovala. A bol Japonec, teda nemal kontakt so západom. Ale maliarstvo malo po dlhý čas fraktálovú stránku. Mohol by som o tom dlho hovoriť. Eiffelova veža má fraktálový aspekt. A čítal som knihu, ktorú o tej veži napísal pán Eiffel. A bolo to naozaj úžasné, ako tomu rozumel.
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.
Toto je chaos, chaos, chaos, Brownova slučka. Jedného dňa som sa rozhodol, že na polceste mojej kariéry som bol zväzovaný toľkými vecami v mojej práci, rozhodol som sa otestovať sám seba. Dokážem sa len tak pozrieť na niečo, na čo všetci pozerali po dlhý čas a nájsť niečo úplne nové? Nuž, pozrel som sa na tieto veci, zvané Brownov pohyb -- iba to chodí dokola. Nejaký čas som sa s tým hral a prinútil som to vrátiť sa na začiatok. Potom som povedal asistentovi, "Nič nevidím. Môžete to nakresliť?" Nakreslil to, čo znamená, že to vyplnil. Povedal: "Nuž, toto z toho vyšlo.." A vravím, "Stačí! Stačí! Stačí! Vidím, je to ostrov." A pekný. Takže Brownov pohyb, ktorý má číslo hrubosti dva, ide dookola. Odmeral som to, 1,33. Znova, znova a znova. Dlhé merania, veľké Brownove pohyby, 1,33. Matematický problém: ako to dokázať? Mojim priateľom to zabralo 20 rokov. Traja z nich mali čiastočné dôkazy. Dali sa dokopy a spolu mali dôkaz. Dostali veľkú (Fieldsovu) medailu za matematiku, jednu z troch medailí, ktoré dostali ľudia dokazujúci veci, ktoré som videl, ale nebol schopný dokázať.
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.
Každý sa ma z času na čas pýta, "Ako to začalo? Čo vás priviedlo k týmto divným témam?" Čo zo mňa spravilo zároveň mechanickýho inžiniera, geografa a matematika a tak ďalej, fyzika? Nuž, vlastne som začal, dosť čudné, študovať ceny na finančnom trhu. A tak, mal som teóriu, napísal som o tom knihy, Prírastky finančných cien. Naľavo vidíte dáta z dlhšieho obdobia. Napravo, hore, vidíte veľmi, veľmi elegantnú teóriu. Bolo to veľmi ľahké a môžete o tom veľmi rýchlo napísať veľa kníh. (Smiech) Sú o tom tisícky kníh. Porovnajte to so skutočnými zmenami ceny. A kde sú skutočné zmeny cien? Nuž, tieto ostatné čiary obsahujú niektoré reálne zmeny cien a pár mojich podvrhov. Takže idea bola, že sa musí dať -- ako sa to povie -- modelovať cenové variácie. A celkom to fungovalo pred 50 rokmi. 50 rokov si ma ľudia doberali, lebo oni to vedeli oveľa oveľa ľahšie. Ale potom ma začali počúvať. (Smiech) Tieto dve krivky sú priemery. Standard & Poor, modrá. A červená je Standard & Poor, bez piatich najväčších nepravidelností. Nepravidelnosti sú otrava. A preto sú v mnohých štúdiách cien, vylučované. "Nuž, Božie zásahy. A zostane vám tá trocha nezmyslov. Božie zásahy." Na tomto obrázku päť Božích zásahov je tak dôležitých ako všetko ostatné. Inými slovami, nie sú to tie Božie zásahy, ktoré by sme mali odložiť. To je podstata, ten problém. Ak zvládnete tie, zvládnete ceny. A ak ich nezvládnete, môžete zvládnuť ten šum ako len chcete, ale bude to jedno. Nuž, tu sú k tomu krivky.
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.
Teraz sa dostávam k hlavnej veci, čo je množina, ktorá nesie moje meno. Svojim spôsobom je to príbeh môjho života. Svoju mladosť som strávil počas nemeckej okupácie Francúzska. A keďže som si myslel, že možno neprežijem deň alebo týždeň, mal som veľké sny. A po vojne, som znova uvidel strýka. Strýko bol slávny matematik a povedal mi, "Pozri, je taký problém, ktorý som nedokázal vyriešiť pred 25 rokmi, a ktorý nikto nevie vyriešiť. Toto je konštrukcia pánov (Gaston) Julia a (Pierre) Fatou. Ak by si našiel niečo nové, hocičo, postavíš si na tom kariéru." Veľmi jednoduché. Tak som sa na to pozrel a ako tisíce ľudí, ktorí to skúšali predtým, nič som nenašiel.
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 potom prišiel počítač. Rozhodol som sa použiť počítač, nie na nové problémy v matematike -- ako tieto čáry-máry, to je nový problém -- ale na staré problémy. A prešiel som od toho, čo voláme reálne čísla, ktoré sú body na priamke, na imaginárne, komplexné čísla, ktoré sú bodmi na rovine, a to sa malo skúsiť. A vyšiel z toho tento tvar. Tento tvar má výnimočnú zložitosť. Rovnica je skrytá tu, z sa rovná z na druhú plus c. Je to také jednoduché, suché. Je to tak nezaujímavé. Otočíte kľukou raz, dvakrát, dvakrát a začnú sa divy. Vypadne z toho toto. Nechcem to vysvetľovať. Vyjde toto. Vyjde toto. Tvary, ktoré sú tak zložité, tak harmonické a také krásne. Toto z toho vychádza opakovane, znova, znova, znova. A to bol jeden z mojich veľkých objavov, objaviť, že tieto ostrovy boli také isté, ako celá tá veľká vec, viac-menej. A potom dostanete všade tieto výnimočné barokové ozdoby. To všetko z tohto malého vzorca, ktorý má, povedzme, päť symbolov. A potom tento. Farba bola pridaná z dvoch dôvodov. Po prvé, lebo tieto tvary sú také zložité, že by z čísel nešlo nič pochopiť. A ak ich nakreslíte, musíte si vybrať nejaký systém. A tak mojim princípom bolo vždy prezentovať tvary rôznym zafarbením, lebo niektoré zafarbenia zdôrazňujú niečo a iné zas to alebo ono. Je to také zložité.
(Laughter)
(Smiech)
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."
V roku 1990 som bol v Cambridge, vo Veľkej Británii, prebrať cenu od univerzity. A o tri dni neskôr pilot prelietal krajinou a našiel túto vec. Odkiaľ sa to tam vzalo? Samozrejme, mimozemšťania. (Smiech) Nuž, noviny v Cambridge uverejnili článok o tomto "náleze" a na ďalší deň dostali 5000 listov od ľudí, čo vraveli, "Ale to je proste veľká Mandelbrotova množina."
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.
Nuž, na záver. Tento tvar vzišiel z čistej matematiky. Bezodné divy prýštia z jednoduchých pravidiel, ktoré sa opakujú bez konca.
Thank you very much.
Ďakujem veľmi pekne.
(Applause)
(Potlesk)