Benoit Mandelbrot: Fractals and the art of roughness

Translator: Roman Studenic Reviewer: Janka Pazurikova

Thank you very much.

Ďakujem veľmi pekne.

Please excuse me for sitting; I'm very old.

Prosím ospravedlňte, že sedím: Som veľmi starý.

(Laughter)

(Smiech)

Well, the topic I'm going to discuss

Nuž, téma o ktorej budem rozprávať,

is one which is, in a certain sense, very peculiar

je v istom zmysle veľmi špecifická,

because it's very old.

pretože je veľmi stará.

Roughness is part of human life

Drsnosť je súčasť ľudského života

forever and forever,

od nepamäti.

and ancient authors have written about it.

Už dávni autori o nej písali.

It was very much uncontrollable,

Je veľmi nekontrolovateľná.

and in a certain sense,

A v istom zmysle

it seemed to be the extreme of complexity,

sa zdalo, že je to extrém zložitosti,

just a mess, a mess and a mess.

proste neporiadok, neporiadok, neporiadok.

There are many different kinds of mess.

Sú rôzne druhy neporiadku.

Now, in fact,

Takže,

by a complete fluke,

úplnou náhodou

I got involved many years ago

som sa dostal veľmi dávno

in a study of this form of complexity,

k štúdiu tejto formy zložitosti.

and to my utter amazement,

A na moje hlboké počudovanie,

I found traces --

som našiel stopy --

very strong traces, I must say --

veľmi silné stopy, musím povedať --

of order in that roughness.

poriadku v tej drsnosti.

And so today, I would like to present to you

A dnes by som vám rád prezentoval

a few examples

zopár príkladov toho,

of what this represents.

čo to predstavuje.

I prefer the word roughness

Preferujem slovo drsnosť

to the word irregularity

pred slovom nepravidelnosť,

because irregularity --

pretože nepravidelnosť --

to someone who had Latin

pre niekoho, kto mal latinčinu,

in my long-past youth --

v mojej dávnej mladosti --

means the contrary of regularity.

znamená opak pravidelnosti.

But it is not so.

Ale tak to nie je.

Regularity is the contrary of roughness

Pravidelnosť je opakom drsnosti,

because the basic aspect of the world

pretože základný aspekt sveta

is very rough.

je veľmi drsný.

So let me show you a few objects.

Ukážem vám pár objektov.

Some of them are artificial.

Niektoré z nich sú umelé.

Others of them are very real, in a certain sense.

Niektoré sú veľmi reálne, v istom zmysle.

Now this is the real. It's a cauliflower.

Toto je ten reálny. Je to karfiol.

Now why do I show a cauliflower,

Ale prečo ukazujem karfiol,

a very ordinary and ancient vegetable?

veľmi obyčajnú a pradávnu zeleninu?

Because old and ancient as it may be,

Pretože tak aký je starý a pradávny,

it's very complicated and it's very simple,

je aj veľmi komplikovaný a veľmi jednoduchý

both at the same time.

zároveň.

If you try to weigh it -- of course it's very easy to weigh it,

Ak ho skúsite odvážiť, samozrejme, je ľahké ho odvážiť.

and when you eat it, the weight matters --

A keď ho jete, záleží na váhe.

but suppose you try to

Ale čo keď skúsite

measure its surface.

zmerať jeho povrch.

Well, it's very interesting.

Nuž, to je veľmi zaujímavé.

If you cut, with a sharp knife,

Ak odrežete ostrým nožom

one of the florets of a cauliflower

jeden kvietok karfiolu

and look at it separately,

a pozriete sa naň samostatne,

you think of a whole cauliflower, but smaller.

vidíte celý karfiol, len menší.

And then you cut again,

A tak ho rozkrojíte znova,

again, again, again, again, again, again, again, again,

znova, znova, znova, znova, znova, znova, znova, znova.

and you still get small cauliflowers.

A stále dostávate malé karfioly.

So the experience of humanity

Zo skúsenosti ľudstva

has always been that there are some shapes

vieme, že sú isté tvary,

which have this peculiar property,

ktoré majú túto zvláštnu vlastnosť,

that each part is like the whole,

že každá časť je ako celok,

but smaller.

len menšia.

Now, what did humanity do with that?

A čo s tým ľudstvo spravilo?

Very, very little.

Veľmi, veľmi málo.

(Laughter)

(Smiech)

So what I did actually is to

Tak som sa do toho pustil,

study this problem,

študoval som tento problém

and I found something quite surprising.

a objavil som niečo dosť prekvapujúce.

That one can measure roughness

Že sa dá merať drsnosť

by a number, a number,

číslom, číslom,

2.3, 1.2 and sometimes much more.

2,3, 1,2 a niekedy oveľa viac.

One day, a friend of mine,

Jedného dňa, jeden priateľ,

to bug me,

aby ma podpichol,

brought a picture and said,

priniesol obrázok a pýta sa,

"What is the roughness of this curve?"

"Aká je drsnosť tejto krivky?"

I said, "Well, just short of 1.5."

Povedal som, "Nuž, tesne pod 1,5"

It was 1.48.

Bolo to 1.48.

Now, it didn't take me any time.

Nezabralo to žiadny čas.

I've been looking at these things for so long.

Už som sa na tieto veci pozeral tak dlho.

So these numbers are the numbers

Tieto čísla sú čísla,

which denote the roughness of these surfaces.

ktoré označujú drsnosť týchto povrchov.

I hasten to say that these surfaces

Rýchlo dodám, že tieto povrchy

are completely artificial.

sú úplne umelé.

They were done on a computer,

Vytvoril ich počítač.

and the only input is a number,

A jediným vstupom je číslo.

and that number is roughness.

A to číslo je drsnosť.

So on the left,

Naľavo,

I took the roughness copied from many landscapes.

som vzal drsnosť prebranú z prírody.

To the right, I took a higher roughness.

Vpravo je vyššia drsnosť.

So the eye, after a while,

A oko, po nejakej dobe,

can distinguish these two very well.

vie tieto dva pomerne dobre rozlíšiť.

Humanity had to learn about measuring roughness.

Ľudstvo sa muselo naučiť merať drsnosť.

This is very rough, and this is sort of smooth, and this perfectly smooth.

Toto je veľmi drsné, toto je pomerne hladké a toho je úplne hladké.

Very few things are very smooth.

Veľmi málo vecí je hladkých.

So then if you try to ask questions:

A potom, ak sa opýtate otázky:

"What's the surface of a cauliflower?"

aký je povrch karfiolu?

Well, you measure and measure and measure.

Nuž, meriate, meriate a meriate.

Each time you're closer, it gets bigger,

Vždy, keď sa priblížite, povrch sa zväčšuje,

down to very, very small distances.

až k veľmi, veľmi malým vzdialenostiam.

What's the length of the coastline

Aká je dĺžka pobrežia

of these lakes?

týchto jazier?

The closer you measure, the longer it is.

Čím z bližšej vzdialenosti sa pozeráte, tým je dlhšia.

The concept of length of coastline,

Koncept dĺžky pobrežia,

which seems to be so natural

čo sa zdá také prirodzené,

because it's given in many cases,

pretože sa udáva na veľa miestach,

is, in fact, complete fallacy; there's no such thing.

je, vlastne, úplná lož; nič také nie je.

You must do it differently.

Musíte to robiť inak.

What good is that, to know these things?

Na čo je to teda dobré, vedieť tieto veci?

Well, surprisingly enough,

Nuž, prekvapujúco,

it's good in many ways.

je to užitočné mnohými spôsobmi.

To begin with, artificial landscapes,

Na začiatok, umelé krajiny,

which I invented sort of,

ktoré som tak trochu vymyslel,

are used in cinema all the time.

sa používajú v kine vlastne stále.

We see mountains in the distance.

Vidíme hory na horizonte.

They may be mountains, but they may be just formulae, just cranked on.

Môžu to byť hory, ale môžu to byť len vzorce, len uletené.

Now it's very easy to do.

Je to vlastne veľmi jednoduché.

It used to be very time-consuming, but now it's nothing.

Bývalo to veľmi časovo náročné, ale dnes je to nič.

Now look at that. That's a real lung.

Pozrite sa na to. To sú skutočné pľúca.

Now a lung is something very strange.

Pľúca sú niečo veľmi čudné.

If you take this thing,

Ak vezmete túto vec,

you know very well it weighs very little.

veľmi dobre viete, koľko váži.

The volume of a lung is very small,

Objem pľúc je veľmi malý.

but what about the area of the lung?

Ale ako je to s povrchom pľúc?

Anatomists were arguing very much about that.

Anatómovia sa o tom dosť sporili.

Some say that a normal male's lung

Niektorí vravia, že normálne mužské pľúca

has an area of the inside

majú povrch

of a basketball [court].

basketbalového ihriska.

And the others say, no, five basketball [courts].

A iní vravia, nie, piatich ihrísk.

Enormous disagreements.

Úžasné nezrovnalosti.

Why so? Because, in fact, the area of the lung

Prečo to? Lebo, v skutočnosti, povrch pľúc,

is something very ill-defined.

je niečo veľmi zle definované.

The bronchi branch, branch, branch

Priedušky sa vetvia, vetvia, vetvia.

and they stop branching,

A potom sa prestanú vetviť,

not because of any matter of principle,

nie kvôli nejakému princípu,

but because of physical considerations:

ale kvôli fyzikálnym okolnostiam,

the mucus, which is in the lung.

kvôli hlienu, ktorý je v pľúcach.

So what happens is that in a way

Takýmto spôsobom

you have a much bigger lung,

máte oveľa väčšie pľúca,

but it branches and branches

ktoré sa vetvia a vetvia,

down to distances about the same for a whale, for a man

až k vzdialenostiam, ktoré sú zhruba rovnaké pre veľryby, pre ľudí

and for a little rodent.

a pre malé hlodavce.

Now, what good is it to have that?

Na čo je to všetko dobré?

Well, surprisingly enough, amazingly enough,

Nuž, prekvapujúco, až udivujúco,

the anatomists had a very poor idea

anatómovia mali až donedávna veľmi chabé znalosti

of the structure of the lung until very recently.

o štruktúre pľúc.

And I think that my mathematics,

A myslím, že moja matematika,

surprisingly enough,

prekvapujúco,

has been of great help

veľmi pomohla

to the surgeons

chirurgom,

studying lung illnesses

ktorí študujú choroby pľúc

and also kidney illnesses,

a tiež ochorenia obličiek,

all these branching systems,

všetkých týchto vetviacich sa systémov,

for which there was no geometry.

pre ktoré nebola žiadna geometria.

So I found myself, in other words,

Inými slovami, ocitol som sa

constructing a geometry,

v konštruovaní geometrie,

a geometry of things which had no geometry.

geometrie vecí, ktoré nemali geometriu.

And a surprising aspect of it

Prekvapujúcim aspektom toho

is that very often, the rules of this geometry

je, že veľmi často, pravidlá tejto geometrie

are extremely short.

sú extrémne krátke.

You have formulas that long.

Máte takéto krátke vzorce.

And you crank it several times.

Párkrát ich pretočíte.

Sometimes repeatedly: again, again, again,

Niekedy opakovane znova, znova, znova.

the same repetition.

To isté opakovanie.

And at the end, you get things like that.

A na koniec dostanete veci ako toto.

This cloud is completely,

Tento oblak je úplne,

100 percent artificial.

100% umelý.

Well, 99.9.

Nuž, 99,9.

And the only part which is natural

Jediná prirodzená časť

is a number, the roughness of the cloud,

je číslo, drsnosť oblaku,

which is taken from nature.

ktoré je prevzaté z prírody.

Something so complicated like a cloud,

Niečo tak komplikované ako oblak,

so unstable, so varying,

tak nestále, tak premenlivé,

should have a simple rule behind it.

by malo mať v pozadí jednoduché pravidlo.

Now this simple rule

Toto jednoduché pravidlo

is not an explanation of clouds.

nie je vysvetlením oblakov.

The seer of clouds had to

Pozorovateľ oblakov to musel

take account of it.

vziať do úvahy.

I don't know how much advanced

Neviem nakoľko pokročilé

these pictures are. They're old.

sú tieto obrázky, sú staré.

I was very much involved in it,

Bol som súčasťou tejto oblasti,

but then turned my attention to other phenomena.

ale potom som obrátil pozornosť na iné fenomény.

Now, here is another thing

Je ďalšia vec,

which is rather interesting.

ktorá je pomerne zaujímavá.

One of the shattering events

Jedna z šokujúcich udalostí

in the history of mathematics,

v histórii matematiky,

which is not appreciated by many people,

nedoceňovaná mnohými ľuďmi,

occurred about 130 years ago,

sa vyskytla asi pred 130 rokmi,

145 years ago.

pred 145 rokmi.

Mathematicians began to create

Matematici začali tvoriť

shapes that didn't exist.

tvary, ktoré neexistovali.

Mathematicians got into self-praise

Matematici začali chváliť sami seba

to an extent which was absolutely amazing,

až do úplne neuveriteľných rozmerov,

that man can invent things

za to, že človek dokáže vymyslieť veci,

that nature did not know.

ktoré príroda nepoznala.

In particular, it could invent

Napríklad, vytvorili

things like a curve which fills the plane.

veci ako krivku, ktorá vyplní rovinu.

A curve's a curve, a plane's a plane,

Krivka je krivka, rovina je rovina

and the two won't mix.

a to nejde dokopy.

Well, they do mix.

Nuž, ide.

A man named Peano

Muž, Peano

did define such curves,

definoval také krivky

and it became an object of extraordinary interest.

a stalo sa to objektom nevídaného záujmu.

It was very important, but mostly interesting

Bolo to veľmi dôležité, ale hlavne zaujímavé,

because a kind of break,

lebo to bol istý druh zlomu,

a separation between

separácia medzi

the mathematics coming from reality, on the one hand,

matematikou vychádzajúcej z reality na jednej strane

and new mathematics coming from pure man's mind.

a novou matematikou, vychádzajúcej z čistej ľudskej mysle.

Well, I was very sorry to point out

Nuž, veľmi nerád som vtedy pripomenul,

that the pure man's mind

že čistá ľudská myseľ

has, in fact,

vlastne,

seen at long last

konečne uvidela,

what had been seen for a long time.

čo bolo k videniu už dlhý čas.

And so here I introduce something,

Takže teraz niečo predstavím,

the set of rivers of a plane-filling curve.

súbor riek krivky vypĺňajúcej rovinu.

And well,

A vlastne,

it's a story unto itself.

je to príbeh sám o sebe.

So it was in 1875 to 1925,

Takže, bolo to v rokoch 1875 až 1925,

an extraordinary period

výnimočné obdobie,

in which mathematics prepared itself to break out from the world.

v ktorej sa matematici pripravovali uniknúť z tohto sveta.

And the objects which were used

A objekty, ktoré používali

as examples, when I was

ako príklady -- keď som bol

a child and a student, as examples

dieťa a študent --

of the break between mathematics

rozchodu medzi matematikou

and visible reality --

a viditeľnou realitou --

those objects,

tieto objekty,

I turned them completely around.

som úplne otočil.

I used them for describing

Použil som ich na popísanie

some of the aspects of the complexity of nature.

niektorých aspektov komplexity prírody.

Well, a man named Hausdorff in 1919

Nuž, istý Hausdorff v roku 1919

introduced a number which was just a mathematical joke,

predstavil číslo, ktoré bolo len matematickým vtipom.

and I found that this number

A prišiel som na to, že toto číslo

was a good measurement of roughness.

je dobrým meradlom drsnosti.

When I first told it to my friends in mathematics

Keď som to prvýkrát povedal priateľom matematikom,

they said, "Don't be silly. It's just something [silly]."

vraveli, "Nebuď blázon. To je len nejaká blbosť."

Well actually, I was not silly.

Nuž, nakoniec, nebola to blbosť.

The great painter Hokusai knew it very well.

Veľký maliar Hokusai to dobre poznal.

The things on the ground are algae.

Tie veci na zemi sú riasy.

He did not know the mathematics; it didn't yet exist.

Nepoznal tú matematiku, ešte neexistovala.

And he was Japanese who had no contact with the West.

A bol Japonec, teda nemal kontakt so západom.

But painting for a long time had a fractal side.

Ale maliarstvo malo po dlhý čas fraktálovú stránku.

I could speak of that for a long time.

Mohol by som o tom dlho hovoriť.

The Eiffel Tower has a fractal aspect.

Eiffelova veža má fraktálový aspekt.

I read the book that Mr. Eiffel wrote about his tower,

A čítal som knihu, ktorú o tej veži napísal pán Eiffel.

and indeed it was astonishing how much he understood.

A bolo to naozaj úžasné, ako tomu rozumel.

This is a mess, mess, mess, Brownian loop.

Toto je chaos, chaos, chaos, Brownova slučka.

One day I decided --

Jedného dňa som sa rozhodol, že

halfway through my career,

na polceste mojej kariéry

I was held by so many things in my work --

som bol zväzovaný toľkými vecami v mojej práci,

I decided to test myself.

rozhodol som sa otestovať sám seba.

Could I just look at something

Dokážem sa len tak pozrieť na niečo,

which everybody had been looking at for a long time

na čo všetci pozerali po dlhý čas

and find something dramatically new?

a nájsť niečo úplne nové?

Well, so I looked at these

Nuž, pozrel som sa na tieto

things called Brownian motion -- just goes around.

veci, zvané Brownov pohyb -- iba to chodí dokola.

I played with it for a while,

Nejaký čas som sa s tým hral

and I made it return to the origin.

a prinútil som to vrátiť sa na začiatok.

Then I was telling my assistant,

Potom som povedal asistentovi,

"I don't see anything. Can you paint it?"

"Nič nevidím. Môžete to nakresliť?"

So he painted it, which means

Nakreslil to, čo znamená,

he put inside everything. He said:

že to vyplnil. Povedal:

"Well, this thing came out ..." And I said, "Stop! Stop! Stop!

"Nuž, toto z toho vyšlo.." A vravím, "Stačí! Stačí! Stačí!

I see; it's an island."

Vidím, je to ostrov."

And amazing.

A pekný.

So Brownian motion, which happens to have

Takže Brownov pohyb, ktorý má

a roughness number of two, goes around.

číslo hrubosti dva, ide dookola.

I measured it, 1.33.

Odmeral som to, 1,33.

Again, again, again.

Znova, znova a znova.

Long measurements, big Brownian motions,

Dlhé merania, veľké Brownove pohyby,

1.33.

1,33.

Mathematical problem: how to prove it?

Matematický problém: ako to dokázať?

It took my friends 20 years.

Mojim priateľom to zabralo 20 rokov.

Three of them were having incomplete proofs.

Traja z nich mali čiastočné dôkazy.

They got together, and together they had the proof.

Dali sa dokopy a spolu mali dôkaz.

So they got the big [Fields] medal in mathematics,

Dostali veľkú (Fieldsovu) medailu za matematiku,

one of the three medals that people have received

jednu z troch medailí, ktoré dostali ľudia

for proving things which I've seen

dokazujúci veci, ktoré som videl,

without being able to prove them.

ale nebol schopný dokázať.

Now everybody asks me at one point or another,

Každý sa ma z času na čas pýta,

"How did it all start?

"Ako to začalo?

What got you in that strange business?"

Čo vás priviedlo k týmto divným témam?"

What got you to be,

Čo zo mňa spravilo

at the same time, a mechanical engineer,

zároveň mechanickýho inžiniera,

a geographer

geografa

and a mathematician and so on, a physicist?

a matematika a tak ďalej, fyzika?

Well actually I started, oddly enough,

Nuž, vlastne som začal, dosť čudné,

studying stock market prices.

študovať ceny na finančnom trhu.

And so here

A tak,

I had this theory,

mal som teóriu,

and I wrote books about it --

napísal som o tom knihy,

financial prices increments.

Prírastky finančných cien.

To the left you see data over a long period.

Naľavo vidíte dáta z dlhšieho obdobia.

To the right, on top,

Napravo, hore,

you see a theory which is very, very fashionable.

vidíte veľmi, veľmi elegantnú teóriu.

It was very easy, and you can write many books very fast about it.

Bolo to veľmi ľahké a môžete o tom veľmi rýchlo napísať veľa kníh.

(Laughter)

(Smiech)

There are thousands of books on that.

Sú o tom tisícky kníh.

Now compare that with real price increments.

Porovnajte to so skutočnými zmenami ceny.

Where are real price increments?

A kde sú skutočné zmeny cien?

Well, these other lines

Nuž, tieto ostatné čiary

include some real price increments

obsahujú niektoré reálne zmeny cien

and some forgery which I did.

a pár mojich podvrhov.

So the idea there was

Takže idea bola,

that one must be able to -- how do you say? --

že sa musí dať -- ako sa to povie --

model price variation.

modelovať cenové variácie.

And it went really well 50 years ago.

A celkom to fungovalo pred 50 rokmi.

For 50 years, people were sort of pooh-poohing me

50 rokov si ma ľudia doberali,

because they could do it much, much easier.

lebo oni to vedeli oveľa oveľa ľahšie.

But I tell you, at this point, people listened to me.

Ale potom ma začali počúvať.

(Laughter)

(Smiech)

These two curves are averages:

Tieto dve krivky sú priemery.

Standard & Poor, the blue one;

Standard & Poor, modrá.

and the red one is Standard & Poor's

A červená je Standard & Poor,

from which the five biggest discontinuities

bez piatich najväčších

are taken out.

nepravidelností.

Now discontinuities are a nuisance,

Nepravidelnosti sú otrava.

so in many studies of prices,

A preto sú v mnohých štúdiách cien,

one puts them aside.

vylučované.

"Well, acts of God.

"Nuž, Božie zásahy.

And you have the little nonsense which is left.

A zostane vám tá trocha nezmyslov.

Acts of God." In this picture,

Božie zásahy." Na tomto obrázku

five acts of God are as important as everything else.

päť Božích zásahov je tak dôležitých ako všetko ostatné.

In other words,

Inými slovami,

it is not acts of God that we should put aside.

nie sú to tie Božie zásahy, ktoré by sme mali odložiť.

That is the meat, the problem.

To je podstata, ten problém.

If you master these, you master price,

Ak zvládnete tie, zvládnete ceny.

and if you don't master these, you can master

A ak ich nezvládnete, môžete zvládnuť

the little noise as well as you can,

ten šum ako len chcete,

but it's not important.

ale bude to jedno.

Well, here are the curves for it.

Nuž, tu sú k tomu krivky.

Now, I get to the final thing, which is the set

Teraz sa dostávam k hlavnej veci, čo je množina,

of which my name is attached.

ktorá nesie moje meno.

In a way, it's the story of my life.

Svojim spôsobom je to príbeh môjho života.

My adolescence was spent

Svoju mladosť som strávil

during the German occupation of France.

počas nemeckej okupácie Francúzska.

Since I thought that I might

A keďže som si myslel, že

vanish within a day or a week,

možno neprežijem deň alebo týždeň,

I had very big dreams.

mal som veľké sny.

And after the war,

A po vojne,

I saw an uncle again.

som znova uvidel strýka.

My uncle was a very prominent mathematician, and he told me,

Strýko bol slávny matematik a povedal mi,

"Look, there's a problem

"Pozri, je taký problém,

which I could not solve 25 years ago,

ktorý som nedokázal vyriešiť pred 25 rokmi,

and which nobody can solve.

a ktorý nikto nevie vyriešiť.

This is a construction of a man named [Gaston] Julia

Toto je konštrukcia pánov (Gaston) Julia

and [Pierre] Fatou.

a (Pierre) Fatou.

If you could

Ak by si našiel

find something new, anything,

niečo nové, hocičo,

you will get your career made."

postavíš si na tom kariéru."

Very simple.

Veľmi jednoduché.

So I looked,

Tak som sa na to pozrel

and like the thousands of people that had tried before,

a ako tisíce ľudí, ktorí to skúšali predtým,

I found nothing.

nič som nenašiel.

But then the computer came,

A potom prišiel počítač.

and I decided to apply the computer,

Rozhodol som sa použiť počítač,

not to new problems in mathematics --

nie na nové problémy v matematike --

like this wiggle wiggle, that's a new problem --

ako tieto čáry-máry, to je nový problém --

but to old problems.

ale na staré problémy.

And I went from what's called

A prešiel som od toho, čo voláme

real numbers, which are points on a line,

reálne čísla, ktoré sú body na priamke,

to imaginary, complex numbers,

na imaginárne, komplexné čísla,

which are points on a plane,

ktoré sú bodmi na rovine,

which is what one should do there,

a to sa malo skúsiť.

and this shape came out.

A vyšiel z toho tento tvar.

This shape is of an extraordinary complication.

Tento tvar má výnimočnú zložitosť.

The equation is hidden there,

Rovnica je skrytá tu,

z goes into z squared, plus c.

z sa rovná z na druhú plus c.

It's so simple, so dry.

Je to také jednoduché, suché.

It's so uninteresting.

Je to tak nezaujímavé.

Now you turn the crank once, twice:

Otočíte kľukou raz, dvakrát,

twice,

dvakrát

marvels come out.

a začnú sa divy.

I mean this comes out.

Vypadne z toho toto.

I don't want to explain these things.

Nechcem to vysvetľovať.

This comes out. This comes out.

Vyjde toto. Vyjde toto.

Shapes which are of such complication,

Tvary, ktoré sú tak zložité,

such harmony and such beauty.

tak harmonické a také krásne.

This comes out

Toto z toho vychádza

repeatedly, again, again, again.

opakovane, znova, znova, znova.

And that was one of my major discoveries,

A to bol jeden z mojich veľkých objavov,

to find that these islands were the same

objaviť, že tieto ostrovy boli také isté,

as the whole big thing, more or less.

ako celá tá veľká vec, viac-menej.

And then you get these

A potom dostanete

extraordinary baroque decorations all over the place.

všade tieto výnimočné barokové ozdoby.

All that from this little formula,

To všetko z tohto malého vzorca,

which has whatever, five symbols in it.

ktorý má, povedzme, päť symbolov.

And then this one.

A potom tento.

The color was added for two reasons.

Farba bola pridaná z dvoch dôvodov.

First of all, because these shapes

Po prvé, lebo tieto tvary

are so complicated

sú také zložité,

that one couldn't make any sense of the numbers.

že by z čísel nešlo nič pochopiť.

And if you plot them, you must choose some system.

A ak ich nakreslíte, musíte si vybrať nejaký systém.

And so my principle has been

A tak mojim princípom bolo

to always present the shapes

vždy prezentovať tvary

with different colorings

rôznym zafarbením,

because some colorings emphasize that,

lebo niektoré zafarbenia zdôrazňujú niečo

and others it is that or that.

a iné zas to alebo ono.

It's so complicated.

Je to také zložité.

(Laughter)

(Smiech)

In 1990, I was in Cambridge, U.K.

V roku 1990 som bol v Cambridge, vo Veľkej Británii,

to receive a prize from the university,

prebrať cenu od univerzity.

and three days later,

A o tri dni neskôr

a pilot was flying over the landscape and found this thing.

pilot prelietal krajinou a našiel túto vec.

So where did this come from?

Odkiaľ sa to tam vzalo?

Obviously, from extraterrestrials.

Samozrejme, mimozemšťania.

(Laughter)

(Smiech)

Well, so the newspaper in Cambridge

Nuž, noviny v Cambridge

published an article about that "discovery"

uverejnili článok o tomto "náleze"

and received the next day

a na ďalší deň dostali

5,000 letters from people saying,

5000 listov od ľudí, čo vraveli,

"But that's simply a Mandelbrot set very big."

"Ale to je proste veľká Mandelbrotova množina."

Well, let me finish.

Nuž, na záver.

This shape here just came

Tento tvar vzišiel

out of an exercise in pure mathematics.

z čistej matematiky.

Bottomless wonders spring from simple rules,

Bezodné divy prýštia z jednoduchých pravidiel,

which are repeated without end.

ktoré sa opakujú bez konca.

Thank you very much.

Ďakujem veľmi pekne.

(Applause)

(Potlesk)