Benoit Mandelbrot: Fractals and the art of roughness

Prevodilac: Lektor: Stevan Radanovic

Thank you very much.

Mnogo vam hvala.

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

Oprostite što sedim, ja sam jako star.

(Laughter)

(Smeh)

Well, the topic I'm going to discuss

Znači, tema o kojoj ću razgovarati

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

je u izvesnom smislu dosta neobična

because it's very old.

jer je vrlo stara.

Roughness is part of human life

Hrapavost je deo ljudskog života,

forever and forever,

i tako će biti za sva vremena.

and ancient authors have written about it.

Još su drevni pisci pisali o tome.

It was very much uncontrollable,

Nju je uglavnom nemoguće kontrolisati.

and in a certain sense,

I u izvesnom smislu,

it seemed to be the extreme of complexity,

čini se da je krajnje složena,

just a mess, a mess and a mess.

samo zbrka i nered.

There are many different kinds of mess.

Postoji mnogo različitih vidova nereda.

Now, in fact,

Zapravo,

by a complete fluke,

zahvaljujući potpuno srećnom slučaju,

I got involved many years ago

pre nekoliko godina sam se uključio

in a study of this form of complexity,

u izučavanje ovog vida kompleksnosti.

and to my utter amazement,

Na moje kranje čuđenje,

I found traces --

pronašao sam tragove –

very strong traces, I must say --

veoma snažne tragove, moram reći –

of order in that roughness.

reda u toj hrapavosti.

And so today, I would like to present to you

Stoga ću danas da vam predstavim

a few examples

nekoliko primera

of what this represents.

onoga što to predstavlja.

I prefer the word roughness

Više mi se sviđa reč hrapavost

to the word irregularity

od nepravilnosti,

because irregularity --

jer nepravilnost –

to someone who had Latin

za nekoga ko je učio latinski

in my long-past youth --

u dalekoj mladosti –

means the contrary of regularity.

znači nešto što je suprotno pravilnosti.

But it is not so.

Ali nije.

Regularity is the contrary of roughness

Pravilnost je suprotnost hrapavosti

because the basic aspect of the world

jer je osnovni aspekt sveta

is very rough.

veoma hrapav.

So let me show you a few objects.

Dopustite mi da vam pokažem nekoliko predmeta.

Some of them are artificial.

Neki od njih su veštački.

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

Neki su stvarni, u izvesnom smislu.

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

Recimo, ovaj je pravi. Ovo je karfiol.

Now why do I show a cauliflower,

Ali zašto pokazujem karfiol,

a very ordinary and ancient vegetable?

jedno jako obično i drevno povrće?

Because old and ancient as it may be,

Zbog toga što, koliko god da je staro i drevno,

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

ono je veoma komplikovano i veoma jednostavno

both at the same time.

u isto vreme.

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

Ako pokušate da mu izmerite težinu, naravno, to je jako jednostavno.

and when you eat it, the weight matters --

A kada ga jedete, težina je bitna.

but suppose you try to

No, pretpostavite da pokušate

measure its surface.

da izmerite njegovu površinu.

Well, it's very interesting.

Da, to je jako zanimljivo.

If you cut, with a sharp knife,

Ako oštrim nožem odsečete

one of the florets of a cauliflower

jedan od njegovih cvetova

and look at it separately,

i pogledate ga odvojeno,

you think of a whole cauliflower, but smaller.

pomislićete na celi karfiol, samo manji.

And then you cut again,

I onda nastavite da režete,

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

dalje, dalje, dalje, dalje, dalje, dalje, dalje, dalje.

and you still get small cauliflowers.

I stalno dobijate male karfiole.

So the experience of humanity

Iskustvo čovečanstva

has always been that there are some shapes

pokazuje da oduvek postoje oblici

which have this peculiar property,

koji poseduju ovakav poseban kvalitet,

that each part is like the whole,

gde svaki deo izgleda kao celina,

but smaller.

samo manja.

Now, what did humanity do with that?

A šta je čovečanstvo učinilo sa time?

Very, very little.

Veoma, veoma malo.

(Laughter)

(Smeh)

So what I did actually is to

Ono što sam ja zapravo učinio

study this problem,

je da sam proučavao ovaj problem

and I found something quite surprising.

i otkrio nešto sasvim iznenađujuće.

That one can measure roughness

Otkrio sam da je moguće izmeriti hrapavost

by a number, a number,

nekim brojem, brojem,

2.3, 1.2 and sometimes much more.

2.3, 1.2, a ponekad i nekim većim.

One day, a friend of mine,

Jedan dan, moj prijatelj je,

to bug me,

kako bi me zavitlavao,

brought a picture and said,

doneo sliku i rekao

"What is the roughness of this curve?"

„Kolika je hrapavost ove obline?“

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

Odgovorio sam, „Pa, nešto manja od 1.5.“

It was 1.48.

Bila je 1.48.

Now, it didn't take me any time.

Znači, uopšte nije bilo potrebno mnogo vremena za to,

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

jer sam toliko dugo posmatrao te stvari.

So these numbers are the numbers

Znači, ovi brojevi su brojevi

which denote the roughness of these surfaces.

koji označavaju hrapavost ovih površina.

I hasten to say that these surfaces

Želim da što pre kažem da su ove površine

are completely artificial.

potpuno veštačke.

They were done on a computer,

Napravljene su kompjuterski.

and the only input is a number,

I jedini unos je broj.

and that number is roughness.

A taj broj je hrapavost.

So on the left,

I tako sam na levoj strani

I took the roughness copied from many landscapes.

uzeo hrapavost preuzetu iz mnogih pejzaža.

To the right, I took a higher roughness.

Na desnoj strani, uzeo sam malo više hrapavosti.

So the eye, after a while,

I tako oko, nakon nekog vremena,

can distinguish these two very well.

može jako dobro napraviti razliku između jednoga i drugoga.

Humanity had to learn about measuring roughness.

Čovečanstvo mora da nauči o merenju hrapavosti.

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

Ovo je hrapavo, ovo je donekle glatko, a ovo je savršeno glatko

Very few things are very smooth.

Izuzetno malo toga je izuzetno glatko.

So then if you try to ask questions:

Pokušajte onda da upitate:

"What's the surface of a cauliflower?"

kolika je površina karfiola?

Well, you measure and measure and measure.

Možete meriti i meriti i meriti.

Each time you're closer, it gets bigger,

Svaki put kada se približite on postane još veći,

down to very, very small distances.

čak i do najmanjih razmaka.

What's the length of the coastline

Kolika je dužina obale

of these lakes?

ovih jezera?

The closer you measure, the longer it is.

Što bliže merite, biće sve duža.

The concept of length of coastline,

Koncept dužine obale,

which seems to be so natural

koja se čini tako prirodnom

because it's given in many cases,

jer se u mnogim slučajevima uzima zdravo za gotovo,

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

zapravo je potpuna obmana; ne postoji ništa takvo.

You must do it differently.

Morate da pokušate na drugačiji način.

What good is that, to know these things?

Kakva je korist znati sve ove stvari?

Well, surprisingly enough,

Hm, dovoljno iznenađujuće,

it's good in many ways.

korisno je na mnogo načina.

To begin with, artificial landscapes,

Za početak, veštački pejzaži

which I invented sort of,

koje sam na neki način izmislio,

are used in cinema all the time.

koriste se u kinematografiji sve vreme.

We see mountains in the distance.

Vidimo planine u daljini.

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

Mogle bi biti planine, ali bi mogle biti i formule, samo sklepane.

Now it's very easy to do.

U današnje vreme je to jako jednostavno učiniti.

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

Nekada je za to bilo potrebno mnogo vremena, ali sada je to sitnica.

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

Pogledajte. Ovo su prava pluća.

Now a lung is something very strange.

Pluća su nešto jako čudno.

If you take this thing,

Ako ih uzmete,

you know very well it weighs very little.

znate jako dobro da je njihova težina mala.

The volume of a lung is very small,

Zapremina pluća je veoma mala.

but what about the area of the lung?

Ali kolika je površina pluća?

Anatomists were arguing very much about that.

Anatomi su se mnogo svađali oko toga.

Some say that a normal male's lung

Neki kažu da pluća normalnog muškarca

has an area of the inside

imaju površinu unutrašnjosti

of a basketball [court].

košarkaškog igrališta.

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

A oni drugi kažu – ne, pet košarkaških igrališta.

Enormous disagreements.

Ogromna razlika u mišljenju.

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

Zbog čega? Zato što je, zapravo, površina pluća

is something very ill-defined.

nešto što je jako loše definisano.

The bronchi branch, branch, branch

Bronhije se razgranavaju, razgranavaju, razgranavaju.

and they stop branching,

I onda prestanu da se granaju,

not because of any matter of principle,

ne zbog bilo kog posebnog principa,

but because of physical considerations:

već zbog fizičkih okolnosti,

the mucus, which is in the lung.

sluzi koja se nalazi u plućima.

So what happens is that in a way

Znači, ono što se dešava jeste da na taj način

you have a much bigger lung,

imate mnogo veća pluća,

but it branches and branches

ali ako se razgranavaju i razgranavaju,

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

sve do rastojanja koja su gotovo jednaka kod kita, čoveka

and for a little rodent.

ili malog glodara.

Now, what good is it to have that?

Ali, kakva je korist od toga?

Well, surprisingly enough, amazingly enough,

Pa, dovoljno iznenađujuće, dovoljno začuđujuće,

the anatomists had a very poor idea

je da su anatomi imali jako bledu ideju

of the structure of the lung until very recently.

o strukturi pluća sve donedavno.

And I think that my mathematics,

A ja mislim da je moja matematika,

surprisingly enough,

sasvim začuđujuće,

has been of great help

bila od velike pomoći

to the surgeons

hirurzima

studying lung illnesses

koji izučavaju bolesti pluća

and also kidney illnesses,

kao i bolesti bubrega,

all these branching systems,

sve te sisteme koji se razgranjavaju,

for which there was no geometry.

za koje nije postojala nikakva geometrija.

So I found myself, in other words,

I tako sam se našao kako, drugim rečima,

constructing a geometry,

konstruišem geometriju,

a geometry of things which had no geometry.

geometriju stvari koje nemaju geometriju.

And a surprising aspect of it

A iznenađujući aspekt toga je

is that very often, the rules of this geometry

da su veoma često pravila ove geometrije

are extremely short.

izuzetno kratka.

You have formulas that long.

Imate formule toliko dugačke.

And you crank it several times.

I onda ih izvršite nekoliko puta.

Sometimes repeatedly: again, again, again,

Ponekad to ponovite više puta.

the same repetition.

Isto ponavljanje.

And at the end, you get things like that.

I na kraju dobijete nešto poput toga.

This cloud is completely,

Ovaj oblak je potpuno,

100 percent artificial.

stoprocentno veštački.

Well, 99.9.

Pa dobro, 99.9.

And the only part which is natural

A jedini deo koji je prirodan

is a number, the roughness of the cloud,

je broj, hrapavost oblaka,

which is taken from nature.

koji je preuzet iz prirode.

Something so complicated like a cloud,

Iza nečega tako komplikovanog poput oblaka,

so unstable, so varying,

tako nestalnog, tako promenljivog,

should have a simple rule behind it.

bi trebalo postojati jednostavno pravilo.

Now this simple rule

Ali to jednostavno pravilo

is not an explanation of clouds.

nije objašnjenje za oblake.

The seer of clouds had to

Posmatrač oblaka mora

take account of it.

to uzeti u obzir.

I don't know how much advanced

Nisam siguran koliko su unapređene

these pictures are. They're old.

ove slike, jer su stare.

I was very much involved in it,

Mnogo sam se bavio njima, .

but then turned my attention to other phenomena.

ali onda sam preusmerio pažnju ne druge fenomene.

Now, here is another thing

Ovde imamo još jednu stvar

which is rather interesting.

koja je dosta zanimljiva.

One of the shattering events

Jedan od dramatičnih događaja

in the history of mathematics,

u istoriji matematike,

which is not appreciated by many people,

a koji veliki broj ljudi uopšte ne ceni,

occurred about 130 years ago,

desio se pre nekih 130,

145 years ago.

145 godina.

Mathematicians began to create

Matematičari su počeli da stvaraju

shapes that didn't exist.

oblike koji nisu postojali.

Mathematicians got into self-praise

Matematičari su počeli sa hvalospevima samim sebi

to an extent which was absolutely amazing,

do te mere gde je potpuno neverovatno

that man can invent things

da čovek može izumeti stvari

that nature did not know.

koje sama priroda nije poznavala.

In particular, it could invent

A posebno da može izumeti

things like a curve which fills the plane.

stvari poput krive koja ispunjava ravan

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

Kriva je kriva, ravan je ravan,

and the two won't mix.

i jedno sa drugim se ne meša.

Well, they do mix.

Ali, ipak se mešaju.

A man named Peano

Čovek po imenu Peano

did define such curves,

je definisao takve krive,

and it became an object of extraordinary interest.

koje su postale predmet neverovatnog interesovanja.

It was very important, but mostly interesting

To je bilo jako bitno, ali uglavnom zanimljivo

because a kind of break,

zbog izvesnog prekida,

a separation between

razdvajanja između

the mathematics coming from reality, on the one hand,

matematike koja proizilazi iz realnosti, s jedne strane

and new mathematics coming from pure man's mind.

i nove matematike koja dolazi iz čistog uma čoveka.

Well, I was very sorry to point out

Bilo mi je jako žao što sam morao ukazati

that the pure man's mind

da je čisti um čoveka

has, in fact,

zapravo

seen at long last

konačno video

what had been seen for a long time.

ono što se moglo videti već dugo vremena.

And so here I introduce something,

I tako sam ovde uveo nešto novo,

the set of rivers of a plane-filling curve.

niz reka krive koja ispunjava ravan.

And well,

Ali,

it's a story unto itself.

to je priča sama za sebe.

So it was in 1875 to 1925,

Dakle, između 1875. i 1925.,

an extraordinary period

u to neverovatno doba

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

matematika se pripremala da se odvoji od sveta.

And the objects which were used

A predmete koji su korišćeni

as examples, when I was

kao primeri, u vreme kada sam ja bio

a child and a student, as examples

dete i student,

of the break between mathematics

primeri prekida između matematike

and visible reality --

i vidljive realnosti –

those objects,

te predmete

I turned them completely around.

sam potpuno preokrenuo.

I used them for describing

Koristio sam ih kako bih opisao

some of the aspects of the complexity of nature.

neke od aspekata složenosti prirode.

Well, a man named Hausdorff in 1919

Čovek po imenu Hausdorf je 1919. godine

introduced a number which was just a mathematical joke,

uveo broj koji je bio samo matematička šala.

and I found that this number

A ja sam otkrio da je taj broj

was a good measurement of roughness.

dobra mera hrapavosti.

When I first told it to my friends in mathematics

Kada sam ga prvi put saopštio mojim prijateljima matematičarima

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

rekli su, "Ne budi lud. To je tek nešto (blesavo)".

Well actually, I was not silly.

No, zapravo, uopšte nije bilo blesavo.

The great painter Hokusai knew it very well.

Čuveni slikar Hokusai je to jako dobro znao.

The things on the ground are algae.

Ovo su alge.

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

On nije bio upoznat sa matematikom; u to vreme još nije postojala.

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

A on je bio Japanac bez ikakvog kontakta sa zapadnim svetom.

But painting for a long time had a fractal side.

No, njegove su slike odavno imale odliku fraktala.

I could speak of that for a long time.

Mogao bih nadugo govoriti o ovome.

The Eiffel Tower has a fractal aspect.

Ajfelov toranj ima fraktalni aspekt.

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

A pročitao sam i knjigu koju je g-din Ajfel napisao o svom tornju.

and indeed it was astonishing how much he understood.

I, zaista, bilo je zapanjujuće koliko mnogo je on razumeo.

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

To je zbrka, zbrka, zbrka, Braunovo kretanje.

One day I decided --

Jednog dana,

halfway through my career,

na pola svoja karijere,

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

toliko toga me je usporavalo u mom poslu,

I decided to test myself.

i odlučio sam da se podvrgnem testu.

Could I just look at something

Da li bih mogao gledati u nešto

which everybody had been looking at for a long time

što svi posmatraju već jako dugo vremena

and find something dramatically new?

i pronaći nešto dramatično novo?

Well, so I looked at these

I tako sam gledao u ove

things called Brownian motion -- just goes around.

stvari koje se nazivaju Braunovim kretanjem – i tako u krug.

I played with it for a while,

Neko vreme sam se igrao sa tim

and I made it return to the origin.

i vratio je na početak.

Then I was telling my assistant,

Govorio sam svome asistentu

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

„Ništa ne vidim. Možeš li ti to naslikati?“

So he painted it, which means

I on je naslikao, što znači

he put inside everything. He said:

da je sve stavio unutra. Rekao je:

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

„Hm, ovo je izašlo....“ A ja sam rekao „Stani! Stani! Stani!

I see; it's an island."

Vidim ga, to je ostrvo."

And amazing.

Zapanjujuće.

So Brownian motion, which happens to have

Znači, Braunovo kretanje, koja ima

a roughness number of two, goes around.

hrapavost broja dva, se okreće.

I measured it, 1.33.

Izmerio sam ga, 1.33.

Again, again, again.

Opet, i opet, i opet.

Long measurements, big Brownian motions,

Duga merenja, velika Braunova kretanja,

1.33.

1.33.

Mathematical problem: how to prove it?

Matematički problem: kako ga dokazati?

It took my friends 20 years.

Mojim prijateljima je trebalo 20 godina.

Three of them were having incomplete proofs.

Trojica njih su imali nepotpune dokaze.

They got together, and together they had the proof.

Oni su se sastali i tek zajedno su imali dokaz.

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

Tako su dobili veliku medalju u matematici,

one of the three medals that people have received

jednu od tri priznanja koja pojedinci primaju

for proving things which I've seen

za dokazivanje stvari koje sam video

without being able to prove them.

ali nisam bio u stanju da dokažem.

Now everybody asks me at one point or another,

I sada me svi pre ili kasnije pitaju

"How did it all start?

„Kako je to sve počelo?

What got you in that strange business?"

Šta te je navelo na taj čudni posao?“

What got you to be,

Šta me je navelo

at the same time, a mechanical engineer,

da istovremeno budem mehanički inženjer,

a geographer

geograf

and a mathematician and so on, a physicist?

i matematičar, i tako dalje, fizičar?

Well actually I started, oddly enough,

Hm, zapravo sam počeo

studying stock market prices.

izučavati cene na berzi.

And so here

I tako sam

I had this theory,

imao ovu teoriju,

and I wrote books about it --

i pisao sam knjige o tome.

financial prices increments.

Rast finansijskih cena.

To the left you see data over a long period.

Na levoj strani vidite podatke prikupljane tokom dugo vremena.

To the right, on top,

Na desnoj, na vrhu,

you see a theory which is very, very fashionable.

vidite teoriju koja je veoma, veoma u modi.

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

Bilo je jako jednostavno i jako brzo možete napisati mnogo knjiga o tome.

(Laughter)

(Smeh)

There are thousands of books on that.

Postoji na hiljade knjiga o tome.

Now compare that with real price increments.

Sada uporedite to sa stvarnim uvećanjem cena

Where are real price increments?

i gde su prava uvećanja cena?

Well, these other lines

Ove druge linije

include some real price increments

uključuju neka realna uvećanja cena,

and some forgery which I did.

i neka lažna, koja sam ja uneo.

So the idea there was

Znači, ideja je bila

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

da je moguće – kako biste to rekli? –

model price variation.

napraviti model varijacije cena.

And it went really well 50 years ago.

I to je išlo jako dobro pre 50 godina.

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

Pedeset godina ljudi su me na neki način omalovažavali

because they could do it much, much easier.

jer su oni to mogli uraditi na mnogo, mnogo jednostavniji način.

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

Ali da vam kažem, ovoga puta, ljudi su me slušali.

(Laughter)

(Smeh)

These two curves are averages:

Ove dve krive su prosečne vrednosti.

Standard & Poor, the blue one;

S&P (agencija za kreditne rejtinge), plave boje.

and the red one is Standard & Poor's

A crvena je S&P,

from which the five biggest discontinuities

iz koje je pet najvećih diskontinuiteta

are taken out.

izdvojeno.

Now discontinuities are a nuisance,

No, diskontinuiteti su smetnja.

so in many studies of prices,

U mnogim studijama cena,

one puts them aside.

oni se ostavljaju po strani.

"Well, acts of God.

"Pa, više sile.

And you have the little nonsense which is left.

I imate malo besmislenost koje je ostalo.

Acts of God." In this picture,

Više sile." Na ovoj slici

five acts of God are as important as everything else.

pet takvih sila su podjednako važne kao i sve ostalo.

In other words,

Drugim rečima,

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

nije viša sila to što bismo trebali staviti po strani.

That is the meat, the problem.

To je suština, problem.

If you master these, you master price,

Ako ovladate njima, ovladali ste cenom.

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

Ako ne ovladate njima, možete ovladati

the little noise as well as you can,

nad ovo malo šuma koliko god želite,

but it's not important.

no to nije važno.

Well, here are the curves for it.

Evo, ovde su krive koje to pokazuju.

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

Sada ću preći na poslednju stvar, a to je niz

of which my name is attached.

od kog je sačinjeno moje ime.

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

Na neki način, to je priča mog života.

My adolescence was spent

Svoje doba adolescencije sam proveo

during the German occupation of France.

tokom nemačke okupacije Francuske.

Since I thought that I might

I s obzirom da sam mislio kako bih

vanish within a day or a week,

mogao nestati kroz dan ili nedelju,

I had very big dreams.

imao sam velike snove.

And after the war,

I nakon rata

I saw an uncle again.

sam ponovo video svog strica.

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

Moj stric je bio jako istaknut matematičar i rekao mi je

"Look, there's a problem

„Pazi, ima jedan problem

which I could not solve 25 years ago,

koji nisam mogao rešiti pre 25 godina,

and which nobody can solve.

i koji niko ne može da reši.

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

To je konstrukcija čoveka po imenu (Gaston) Julia

and [Pierre] Fatou.

i (Pierre) Fatou.

If you could

Ako bi uspeo

find something new, anything,

da nađeš nešto novo, bilo šta,

you will get your career made."

napravićeš karijeru“.

Very simple.

Veoma jednostavno.

So I looked,

I tako sam počeo tražiti,

and like the thousands of people that had tried before,

i kao i hiljade onih koji su pokušali pre mene,

I found nothing.

nisam ništa našao.

But then the computer came,

A onda se pojavio kompjuter.

and I decided to apply the computer,

I ja sam odlučio da primenim kompjuter,

not to new problems in mathematics --

ne na nove matematičke probleme –

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

poput ovog mrdanja ovde, to je novi problem –

but to old problems.

već na stare probleme.

And I went from what's called

I krenuo sam od onoga što se zove

real numbers, which are points on a line,

realni brojevi, koji predstavljaju tačke na liniji,

to imaginary, complex numbers,

ka imaginarnim, složenim brojevima,

which are points on a plane,

koji su tačke na ravni,

which is what one should do there,

i to je ono što treba da se tu uradi.

and this shape came out.

I dobio sam ovaj oblik.

This shape is of an extraordinary complication.

Ovaj oblik je neverovatno komplikovan.

The equation is hidden there,

Ovde je skrivena jednačina,

z goes into z squared, plus c.

z ide u z na kvadrat, plus c.

It's so simple, so dry.

Jako je jednostavno, jako suvo.

It's so uninteresting.

I tako nezanimljivo.

Now you turn the crank once, twice:

Okrenite ovaj kotur jednom, drugi put,

twice,

još jednom,

marvels come out.

i izaći će čudo.

I mean this comes out.

Mislim, ovo će izaći.

I don't want to explain these things.

Ne želim da objašnjavam ove stvari.

This comes out. This comes out.

Pojavljuje se ovo. I ovo.

Shapes which are of such complication,

Oblici koji su tako komplikovani,

such harmony and such beauty.

takvog sklada i lepote.

This comes out

Ovo izlazi

repeatedly, again, again, again.

ponovo, iznova i iznova.

And that was one of my major discoveries,

I to je bilo jedno od mojih glavnih otkrića

to find that these islands were the same

da su ova ostrva ista

as the whole big thing, more or less.

kao i cela velika stvar, manje ili više.

And then you get these

A onda imate ove

extraordinary baroque decorations all over the place.

neobične barokne dekoracije posvuda.

All that from this little formula,

I sve to iz ove male formule,

which has whatever, five symbols in it.

koja se sastoji od koliko god, nekih pet simbola.

And then this one.

A potom ova.

The color was added for two reasons.

Boja je dodana iz dva razloga.

First of all, because these shapes

Najpre zbog toga što su ovi oblici

are so complicated

previše komplikovani

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

da bi iko mogao videti bilo kakav smisao u brojevima.

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

A ako ih unosite u grafikon, morate odabrati neki sistem.

And so my principle has been

A moj princip je oduvek bio

to always present the shapes

da predstavim oblike

with different colorings

različitim bojama,

because some colorings emphasize that,

zato što neke boje naglašavaju ovo,

and others it is that or that.

a neke druge ovo ili ono.

It's so complicated.

Tako je komplikovano.

(Laughter)

(Smeh)

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

1990. godine sam bio u Kembridžu, u Ujedinjenom Kraljevstvu

to receive a prize from the university,

kako bih primio nagradu univerziteta.

and three days later,

I tri dana nakon toga,

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

jedan pilot je leteo iznad pejzaža i otkrio ovo.

So where did this come from?

Odakle se ovo pojavilo?

Obviously, from extraterrestrials.

Očigledno, od vanzemaljaca.

(Laughter)

(Smeh)

Well, so the newspaper in Cambridge

Tako su novine u Kembridžu

published an article about that "discovery"

objavile članak o „otkriću“

and received the next day

i sledeći dan primile

5,000 letters from people saying,

5.000 pisama od čitalaca koji su rekli

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

„Ali to je naprosto veoma veliki Mandelbrotov set“.

Well, let me finish.

Dopustite mi da završim.

This shape here just came

Ovaj oblik ovde je nastao

out of an exercise in pure mathematics.

kao vežba u čistoj matematici.

Bottomless wonders spring from simple rules,

Beskrajna čuda izviru iz jednostavnih pravila,

which are repeated without end.

koja se beskrajno ponavljaju.

Thank you very much.

Hvala vam mnogo.

(Applause)

Aplauz.