Benoit Mandelbrot: Fractals and the art of roughness

Prevoditelj: Sanja Srbljnović Čuček Recezent: Neda Vrkic

Puno vam hvala.

Oprostite što sjedim; vrlo sam star.

(Smijeh)

Dakle, tema o kojoj ću govoriti je u određenom smislu, vrlo neobična

Thank you very much.

jer je vrlo stara.

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

Hrapavost je od pamtivijeka dio ljudskog života

(Laughter)

Well, the topic I'm going to discuss

i drevni su pisci pisali o njoj.

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

Nije ju bilo moguće kontrolirati,

because it's very old.

i na neki način,

Roughness is part of human life

čini se krajnje složena, zbrkana i zbrčkana.

forever and forever,

and ancient authors have written about it.

Mnogo je vrsta zbrkanosti.

Zapravo, pukim slučajem,

It was very much uncontrollable,

and in a certain sense,

pred mnogo godina, bio sam uključen

it seemed to be the extreme of complexity,

u studiju tog oblika kompleksnosti,

just a mess, a mess and a mess.

There are many different kinds of mess.

i na moje veliko iznenađenje,

Now, in fact,

našao sam tragove --

by a complete fluke,

vrlo jasne tragove, moram reći --

I got involved many years ago

reda u toj hrapavosti.

in a study of this form of complexity,

Tako vam danas želim dati

nekoliko primjera onog što to predstavlja.

and to my utter amazement,

I found traces --

Draža mi je riječ hrapavost

very strong traces, I must say --

nego riječ nepravilnost

of order in that roughness.

jer nepravilnost --

And so today, I would like to present to you

nekome tko je imao latinski

a few examples

u svojoj dalekoj mladosti --

of what this represents.

znači suprotno od pravilnosti.

I prefer the word roughness

Ali to nije tako.

to the word irregularity

Pravilnost je suprotna hrapavosti

because irregularity --

to someone who had Latin

jer je osnovni aspekt svijeta vrlo hrapav.

in my long-past youth --

means the contrary of regularity.

Da vam pokažem nekoliko predmeta.

But it is not so.

Neki od njih su umjetni.

Regularity is the contrary of roughness

Drugi od njih su vrlo stvarni, u nekom smislu.

because the basic aspect of the world

Ovo je stvarno. To je šenon.

is very rough.

A zašto vam pokazujem šenon,

So let me show you a few objects.

tako obično i drevno povrće?

Some of them are artificial.

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

Jer, koliko god da bilo staro i drevno,

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

vrlo je komplicirano

i vrlo jednostavno,

Now why do I show a cauliflower,

sve u isti mah.

Važete li ga -- naravno, lako ga je izvagati,

a very ordinary and ancient vegetable?

a kad ga jedete, težina je važna --

Because old and ancient as it may be,

no pretpostavimo da pokušate izmjeriti njegovu površinu.

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,

Dakle, vrlo je zanimljivo.

Odrežete li oštrim nožem

and when you eat it, the weight matters --

jednu od cvjetnih grana šenona

but suppose you try to

i pogledate je zasebno,

mislit ćete na cijeli šenon, ali manji.

measure its surface.

Well, it's very interesting.

A onda opet odrežete i opet i opet i opet i opet i opet

If you cut, with a sharp knife,

i još uvijek dobivate male šenone.

one of the florets of a cauliflower

and look at it separately,

Ljudskom je iskustvu dakle oduvijek poznato

you think of a whole cauliflower, but smaller.

da postoje neki oblici s tim posebnim svojstvom,

And then you cut again,

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

da je svaki dio poput cjeline,

and you still get small cauliflowers.

ali manji.

So the experience of humanity

A što je čovječanstvo s time napravilo?

has always been that there are some shapes

Vrlo, vrlo malo.

which have this peculiar property,

(Smijeh)

that each part is like the whole,

I što sam zapravo učinio -

but smaller.

proučavao sam ovaj problem

Now, what did humanity do with that?

Very, very little.

i otkrio nešto prilično iznenađujuće.

(Laughter)

Da se hrapavost može mjeriti nekim brojem,

So what I did actually is to

study this problem,

2,3; 1,2 a katkad i nekim većim.

and I found something quite surprising.

Jednom mi je neki prijatelj, da me zagnjavi,

That one can measure roughness

donio neku sliku i pitao,

“Kolika je hrapavost ove krivulje?”

by a number, a number,

Rekao sam, “Nešto manja od 1,5.”

2.3, 1.2 and sometimes much more.

Bila je 1,48.

One day, a friend of mine,

Dakle, nije mi trebalo puno vremena.

to bug me,

Dugo sam proučavao te stvari.

brought a picture and said,

To su dakle brojevi koji mogu opisati hrapavost tih površina.

"What is the roughness of this curve?"

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

It was 1.48.

Moram vam reći da su te površine potpuno umjetne.

Now, it didn't take me any time.

Napravljene su računalom,

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

a jedini je unos bio broj,

So these numbers are the numbers

a taj je broj hrapavost.

which denote the roughness of these surfaces.

Tako sam na lijevoj slici

I hasten to say that these surfaces

uzeo hrapavost kopiranu s mnogih pejzaža.

are completely artificial.

They were done on a computer,

Na desnoj sam uzeo veću hrapavost.

and the only input is a number,

Tako oko, nakon nekog vremena

and that number is roughness.

može dobro razlikovati jednu od druge.

So on the left,

Čovječanstvo je moralo učiti mjeriti hrapavost.

I took the roughness copied from many landscapes.

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

To the right, I took a higher roughness.

Vrlo je malo stvari vrlo glatko.

So the eye, after a while,

Pa ako pokušate s pitanjima:

can distinguish these two very well.

Humanity had to learn about measuring roughness.

“Kolika je površina šenona?”

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

Mjerit ćete i mjeriti i mjeriti.

Very few things are very smooth.

Kad god se približite, postaje veća,

sve do vrlo, vrlo malih razmaka.

So then if you try to ask questions:

Kolika je duljina obalne crte ovih jezera?

"What's the surface of a cauliflower?"

Well, you measure and measure and measure.

Što bliže mjerite, to je dulja.

Each time you're closer, it gets bigger,

Koncept duljine obalne crte,

koji se čini tako prirodan, jer se navodi na mnogo mjesta,

down to very, very small distances.

What's the length of the coastline

je zapravo zabluda; tako nešto ne postoji.

of these lakes?

The closer you measure, the longer it is.

Morate tome drugačije pristupiti.

The concept of length of coastline,

Što će vam to znanje?

which seems to be so natural

Pa, nećete vjerovati,

because it's given in many cases,

korisno je na mnogo načina.

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

Za početak, umjetni pejzaži, koje sam na neki način smislio,

You must do it differently.

koriste se u kinematografiji sve vrijeme.

What good is that, to know these things?

Vidimo planine u daljini.

Well, surprisingly enough,

To mogu biti planine, a mogu biti i samo pokrenute formule.

it's good in many ways.

To begin with, artificial landscapes,

Sad je to vrlo lako učiniti.

which I invented sort of,

Nekad je to bilo vrlo dugotrajno, sad to nije ništa.

are used in cinema all the time.

Pogledajte ovo. To su prava pluća.

We see mountains in the distance.

Pluća su nešto vrlo čudnovato.

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

Uzmete li ih u ruke, znat ćete da vrlo malo teže.

Now it's very easy to do.

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

Volumen pluća je vrlo malen,

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

no što je s površinom pluća?

Now a lung is something very strange.

Anatomi su o tome puno raspravljali.

If you take this thing,

Neki kažu da normalna muška pluća

you know very well it weighs very little.

The volume of a lung is very small,

imaju površinu unutrašnjeg košarkaškog igrališta.

but what about the area of the lung?

A drugi kažu, ne, pet takvih igrališta.

Anatomists were arguing very much about that.

Poprilično neslaganje.

Some say that a normal male's lung

A zašto?

Zato što je površina pluća

has an area of the inside

nešto što je vrlo loše definirano.

of a basketball [court].

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

Bronhiji se granaju, granaju, granaju

i prestaju se granati, ne zbog nekog principa,

Enormous disagreements.

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

is something very ill-defined.

već iz fizičkog razloga: zbog sluzi koja je u plućima.

The bronchi branch, branch, branch

and they stop branching,

Pa se događa da možete imati

i daleko veća pluća,

not because of any matter of principle,

no ona će se granati i granati do otprilike istih distanci

but because of physical considerations:

i kod kita i kod čovjeka, kao i kod sitnog glodavca.

the mucus, which is in the lung.

A kakve koristi od toga?

So what happens is that in a way

you have a much bigger lung,

Pa začudo, nećete vjerovati,

but it branches and branches

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

anatomi su imali vrlo slabu predodžbu strukture pluća, sve donedavno.

and for a little rodent.

I mislim da je moja matematika,

Now, what good is it to have that?

za divno čudo,

Well, surprisingly enough, amazingly enough,

bila od velike pomoći kirurzima

the anatomists had a very poor idea

koji su proučavali plućne bolesti

of the structure of the lung until very recently.

kao i bolesti bubrega

And I think that my mathematics,

i svih sustava koji se granaju,

surprisingly enough,

a za koje nije bilo geometrije.

has been of great help

Tako sam se takoreći našao u konstruiranju geometrije,

to the surgeons

studying lung illnesses

geometrije stvari koje nisu imale geometriju.

and also kidney illnesses,

all these branching systems,

Tu je začuđujuće da su vrlo često

for which there was no geometry.

pravila te geometrije izuzetno kratka.

So I found myself, in other words,

constructing a geometry,

Imate ovoliko duge formule.

a geometry of things which had no geometry.

Zavrtite ih nekoliko puta.

Ponekad ponavljano: opet, opet i opet isto ponavljanje.

And a surprising aspect of it

is that very often, the rules of this geometry

Na kraju, dobijete nešto poput ovog.

are extremely short.

Ovaj oblak je u potpunosti, sto posto umjetan.

You have formulas that long.

And you crank it several times.

Dobro, 99,9.

Sometimes repeatedly: again, again, again,

A jedini prirodni dio je broj,

the same repetition.

And at the end, you get things like that.

hrapavost tog oblaka, koji je uzet iz prirode.

This cloud is completely,

100 percent artificial.

Nešto toliko komplicirano kao oblak,

tako nestabilno, tako varirajuće,

Well, 99.9.

treba nositi u sebi jednostavno pravilo.

And the only part which is natural

is a number, the roughness of the cloud,

A to jednostavno pravilo

which is taken from nature.

nije objašnjenje oblaka.

Something so complicated like a cloud,

Promatrač oblaka mora to uzeti u obzir.

so unstable, so varying,

should have a simple rule behind it.

Ne znam koliko su napredne ove slike -- stare su.

Now this simple rule

is not an explanation of clouds.

Prilično sam se bavio time,

a onda pažnju posvetio drugim fenomenima.

The seer of clouds had to

take account of it.

A ima još jedna prilično interesantna stvar.

I don't know how much advanced

these pictures are. They're old.

Jedan od dramatičnih događaja u povijesti matematike,

I was very much involved in it,

but then turned my attention to other phenomena.

kojeg mnogo ljudi ne razumije,

Now, here is another thing

dogodio se prije oko 130, 145 godina.

which is rather interesting.

Matematičari su počeli kreirati oblike koji nisu postojali.

One of the shattering events

in the history of mathematics,

Matematičari su bili puni samohvale

which is not appreciated by many people,

do mjere koja je bila zapanjujuća,

occurred about 130 years ago,

da čovjek može smisliti stvari koje priroda nije poznavala.

145 years ago.

Mathematicians began to create

Točnije, mogao je smisliti nešto poput krivulje koja ispunjava ravninu.

shapes that didn't exist.

Mathematicians got into self-praise

Krivulja je krivulja, a ravnina ravnina,

to an extent which was absolutely amazing,

i to dvoje se ne želi miješati.

that man can invent things

A ipak se miješaju.

that nature did not know.

Čovjek po imenu Peano definirao je takve krivulje,

In particular, it could invent

things like a curve which fills the plane.

što je postalo predmetom nevjerojatnog zanimanja.

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

and the two won't mix.

Bilo je to vrlo važno, no uglavnom zanimljivo

Well, they do mix.

zbog svojevrsnog raskola,

A man named Peano

razdvajanja matematike koja dolazi iz stvarnosti,

did define such curves,

and it became an object of extraordinary interest.

i nove matematike koja dolazi iz čistog ljudskog uma.

It was very important, but mostly interesting

Nažalost sam pokazao

because a kind of break,

da je čisti ljudski um ustvari konačno uvidio

a separation between

the mathematics coming from reality, on the one hand,

and new mathematics coming from pure man's mind.

ono što je već odavno bilo viđeno.

Tako ovdje uvodim nešto,

Well, I was very sorry to point out

niz rijeka krivulje koja ispunjava ravninu.

that the pure man's mind

has, in fact,

No dobro, to je priča za sebe.

seen at long last

what had been seen for a long time.

Bilo je to od 1875. do 1925.,

And so here I introduce something,

u fascinantnom razdoblju

the set of rivers of a plane-filling curve.

u kojem se matematika pripremala za odvajanje od svijeta.

And well,

A predmeti korišteni dok sam bio đak i student

it's a story unto itself.

So it was in 1875 to 1925,

kao primjeri raskola matematike i vidljive realnosti --

an extraordinary period

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

te predmete, potpuno sam ih obrnuo.

And the objects which were used

as examples, when I was

Koristio sam ih za opisivanje nekih aspekata kompleksnosti prirode.

a child and a student, as examples

of the break between mathematics

and visible reality --

1919., čovjek imenom Hausdorff

those objects,

uveo je broj koji je smatran samo matematičkom šalom,

I turned them completely around.

I used them for describing

a ja sam otkrio da je taj broj dobra mjera za hrapavost.

some of the aspects of the complexity of nature.

Kad su to moji matematičari čuli, rekli su,

Well, a man named Hausdorff in 1919

“Ne budi blesav, to je samo...”

introduced a number which was just a mathematical joke,

Pa zapravo, nisam bio blesav.

and I found that this number

Slavni slikar Hokusai to je dobro znao.

was a good measurement of roughness.

Ono na tlu su alge.

When I first told it to my friends in mathematics

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

On nije poznavao tu matematiku; toga još nije bilo.

A on je bio Japanac koji nije imao kontakta sa Zapadom.

Well actually, I was not silly.

No slikarstvo je odavna imalo fraktalnu stranu.

The great painter Hokusai knew it very well.

O tome bih mogao dugo pričati.

The things on the ground are algae.

Eiffelov toranj ima fraktalni aspekt.

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

Pročitao sam knjigu g. Eiffela o njegovom tornju,

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

i zaista je fascinantno koliko je toga shvaćao.

But painting for a long time had a fractal side.

I could speak of that for a long time.

To je zbrka, Brownova petlja. [Brownov klaster]

The Eiffel Tower has a fractal aspect.

Jednog dana sam odlučio --

negdje na polovici svoje karijere.

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

Bio sam ograničen s puno stvari u poslu --

and indeed it was astonishing how much he understood.

odlučio sam se iskušati.

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

Bi li mogao samo gledati nešto

što su već svi dugo vremena promatrali

One day I decided --

halfway through my career,

i pronaći nešto geometrijski novo?

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

Tako sam promatrao ove stvari

I decided to test myself.

Could I just look at something

nazvane Brownovo gibanje -- samo ide uokolo.

which everybody had been looking at for a long time

Neko sam se vrijeme igrao njime,

and find something dramatically new?

i natjerao ga da se vrati na početak.

Onda bih rekao svom asistentu,

Well, so I looked at these

“Ne vidim ništa; možeš li to nacrtati?”

things called Brownian motion -- just goes around.

I on je to nacrtao, sve je stavio unutra.

I played with it for a while,

Rekao je: “Pa ispalo je ovo...”

and I made it return to the origin.

Povikao sam, “Stani, stani, stani!

Vidim, to je otok.”

Then I was telling my assistant,

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

Upravo nevjerojatno.

So he painted it, which means

Dakle, Brownovo gibanje,

koje ima hrapavost broja dva, ide uokolo.

he put inside everything. He said:

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

Izmjerio sam ga, 1,33.

I see; it's an island."

Opet, i opet, i opet.

Duga mjerenja, velika Brownova gibanja, 1,33.

And amazing.

So Brownian motion, which happens to have

Matematički problem: kako to dokazati?

a roughness number of two, goes around.

Mojim je prijateljima trebalo 20 godina.

I measured it, 1.33.

Again, again, again.

Trojica su imala nepotpune dokaze.

Long measurements, big Brownian motions,

Sastali su se i zajedno našli dokaz.

1.33.

Mathematical problem: how to prove it?

Pa su dobili veliku [Fieldsovu] medalju u matematici,

It took my friends 20 years.

jednu od tri medalje koju su drugi dobili

Three of them were having incomplete proofs.

jer su dokazali stvari koje sam vidio

They got together, and together they had the proof.

ali ih nisam mogao dokazati.

Svi me prije ili kasnije pitaju,

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

“Kako je to sve započelo?

one of the three medals that people have received

Što vas je dovelo u te neobične vode?

for proving things which I've seen

without being able to prove them.

Što sve morate istovremeno biti -

strojarski inženjer, geograf, matematičar i fizičar?”

Now everybody asks me at one point or another,

"How did it all start?

What got you in that strange business?"

Zapravo sam počeo, nećete vjerovati, proučavajući cijene na burzi.

What got you to be,

I tako sam tu razvio ovu teoriju i o njoj pisao knjige --

at the same time, a mechanical engineer,

a geographer

and a mathematician and so on, a physicist?

Well actually I started, oddly enough,

rast financijskih cijena.

Lijevo su podaci prikupljani dugo vremena.

studying stock market prices.

And so here

Na vrhu zdesna,

vidite teoriju koja je vrlo, vrlo moderna.

I had this theory,

Jednostavna je i možete o njoj vrlo brzo napisati puno knjiga.

and I wrote books about it --

financial prices increments.

(Smijeh)

To the left you see data over a long period.

Na tisuće je knjiga napisano o tome.

To the right, on top,

Sad to usporedite sa stvarnim pomacima cijena.

you see a theory which is very, very fashionable.

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

Gdje su stvarni pomaci cijena?

Dakle, ove druge linije uključuju neke stvarne pomake cijena

(Laughter)

There are thousands of books on that.

i neke lažne koje sam ja kreirao.

Tako je tu ideja bila da se mora moći --

Now compare that with real price increments.

Where are real price increments?

kako bi to rekli? -- dati model varijacije cijena.

Well, these other lines

I to je stvarno dobro išlo pred 50 godina.

include some real price increments

and some forgery which I did.

Čitavih 50 godina su me pomalo omalovažavali

So the idea there was

jer se to moglo puno lakše izvesti.

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

Ali kažem vam, tada su me ljudi slušali.

model price variation.

And it went really well 50 years ago.

(Smijeh)

Ove dvije krivulje su prosječne vrijednosti:

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

[indeks agencije] S&P, plava,

because they could do it much, much easier.

a crvena [indeks agencije] S&P bez pet najvećih diskontinuiteta.

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

(Laughter)

Diskontinuiteti su gnjavaža

These two curves are averages:

Standard & Poor, the blue one;

pa se u mnogim analizama cijena oni izostavljaju.

and the red one is Standard & Poor's

from which the five biggest discontinuities

“Dakle, viša sila.”

A ono što ostaje je malo besmisleno.

are taken out.

Now discontinuities are a nuisance,

Na ovoj je slici pet viših sila važno koliko i sve ostalo.

so in many studies of prices,

one puts them aside.

"Well, acts of God.

Drugim riječima, višu silu ne bi trebali držati po strani.

And you have the little nonsense which is left.

To je srž, to je problem.

Acts of God." In this picture,

Ovladate li time, ovladat ćete cijenom,

five acts of God are as important as everything else.

a ne ovladate li time,

možete se do mile volje uspješno baviti šumom,

In other words,

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

ali to nije važno.

That is the meat, the problem.

Dakle, tu su krivulje za to.

Sad stižem do konačne stvari,

If you master these, you master price,

a to je set koji nosi moje ime.

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

Na neki način, moja životna priča.

the little noise as well as you can,

Bio sam adolescent za njemačke okupacije Francuske.

but it's not important.

Well, here are the curves for it.

Budući da sam mislio da bih mogao nestati za koji dan ili tjedan,

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.

imao sam vrlo velike snove.

My adolescence was spent

A nakon rata,

during the German occupation of France.

ponovno sam vidio ujaka.

Since I thought that I might

Moj mi je ujak, čuveni matematičar, rekao,

vanish within a day or a week,

“Gle, postoji problem koji nisam uspio riješiti prije 25 godina

I had very big dreams.

i nitko ga ne može riješiti.

And after the war,

To je konstrukcija matematičara [Gaston] Julia i [Pierre] Fatou.

I saw an uncle again.

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

Kad bi mogao otkriti nešto novo, bilo što,

"Look, there's a problem

which I could not solve 25 years ago,

napravio bi si karijeru.”

and which nobody can solve.

Vrlo jednostavno.

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

Proučavao sam to, i poput tisuća njih prije mene,

and [Pierre] Fatou.

If you could

nisam našao ništa.

find something new, anything,

A onda se pojavilo računalo

you will get your career made."

pa sam odlučio primijeniti računalo,

Very simple.

So I looked,

ne za nove probleme u matematici --

and like the thousands of people that had tried before,

kao ovo miganje, to je novi problem --

I found nothing.

već za stare probleme.

I pošao sam od onog što se naziva realni brojevi, što su točke na pravcu,

But then the computer came,

and I decided to apply the computer,

prema imaginarnim, kompleksnim brojevima, što su točke u ravnini,

not to new problems in mathematics --

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

a to je ono što bi se trebalo raditi,

but to old problems.

i pojavio se ovaj oblik.

And I went from what's called

Ovaj je oblik izuzetno složen.

real numbers, which are points on a line,

to imaginary, complex numbers,

Jednadžba koja se tu skriva je z prelazi u z na kvadrat, plus c.

which are points on a plane,

which is what one should do there,

Tako jednostavno, tako suhoparno.

and this shape came out.

Tako nezanimljivo.

This shape is of an extraordinary complication.

Sad je izvrtite jednom, dvaput:

The equation is hidden there,

dvaput -- čudesa nastaju.

z goes into z squared, plus c.

It's so simple, so dry.

Mislim, nastaje ovo.

Ne želim objašnjavati te stvari.

It's so uninteresting.

Now you turn the crank once, twice:

Nastaje ovo, i ovo.

Oblici tako zamršeni, a tako skladni i lijepi.

twice,

marvels come out.

Ovo nastaje ponavljano, opet, opet i opet.

I mean this comes out.

I don't want to explain these things.

A to je bilo jedno od mojih velikih otkrića

This comes out. This comes out.

da su ti otoci bili isti kao i čitava cjelina, manje-više.

Shapes which are of such complication,

such harmony and such beauty.

A onda posvuda dobijete ove neobične barokne uzorke.

This comes out

repeatedly, again, again, again.

And that was one of my major discoveries,

I sve to iz ove male formule,

to find that these islands were the same

koja ima nekoliko, pet simbola, u sebi.

as the whole big thing, more or less.

And then you get these

A onda ovo.

Boja je dodana iz dva razloga.

extraordinary baroque decorations all over the place.

Prvo, jer su ti oblici toliko složeni

All that from this little formula,

which has whatever, five symbols in it.

da se ne bi mogao uhvatiti smisao tih brojeva.

And then this one.

A ako ih crtate ploterom, morate izabrati neki sistem.

The color was added for two reasons.

Tako je moj princip bio

First of all, because these shapes

are so complicated

uvijek prikazati oblike u različitim bojama

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

jer neke boje naglašavaju ovo, a druge ono.

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

To je tako zamršeno.

And so my principle has been

(Smijeh)

to always present the shapes

1990., bio sam u Cambridgeu, U.K.

with different colorings

i primao nagradu tog sveučilišta,

because some colorings emphasize that,

a tri dana kasnije,

and others it is that or that.

pilot je nadlijetao područje i našao ovo.

It's so complicated.

(Laughter)

Odakle je to došlo?

Očito, od izvanzemaljaca.

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

to receive a prize from the university,

(Smijeh)

and three days later,

Dakle, novine u Cambridgeu objavile su članak o tom “otkriću”

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

i sljedećeg dana primile 5000 pisama u kojima je pisalo

So where did this come from?

Obviously, from extraterrestrials.

“Ma to je samo vrlo velik Mandelbrotov set.”

(Laughter)

Well, so the newspaper in Cambridge

Dakle, da završim.

published an article about that "discovery"

Ovaj je oblik proizašao iz vježbanja čiste matematike.

and received the next day

5,000 letters from people saying,

Neizmjerna čuda proizlaze iz jednostavnih pravila,

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

što se ponavljaju bez kraja.

Well, let me finish.

Najljepša vam hvala.

This shape here just came

(Pljesak)

out of an exercise in pure mathematics.

Bottomless wonders spring from simple rules,

which are repeated without end.

Thank you very much.

(Applause)