TED2010
Benoit Mandelbrot: Fractals and the art of roughness
Benoa Mandelbrot: Fraktali i umetnost hrapavosti
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Na TED2010, matematička legenda Benoa Mandelbrot razvija temu o kojoj je prvi put pričao na TED1984 – o izuzetnoj kompleksnosti hrapavosti i načinu na koji matematika fraktala može da pronađe red unutar obrazaca koji se čine nepojmljivo komplikovanim.
Benoit Mandelbrot - Mathematician
Benoit Mandelbrot's work led the world to a deeper understanding of fractals, a broad and powerful tool in the study of roughness, both in nature and in humanity's works. Full bio
Benoit Mandelbrot's work led the world to a deeper understanding of fractals, a broad and powerful tool in the study of roughness, both in nature and in humanity's works. Full bio
Double-click the English transcript below to play the video.
00:15
Thank you very much.
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Mnogo vam hvala.
00:17
Please excuse me for sitting; I'm very old.
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Oprostite što sedim, ja sam jako star.
00:20
(Laughter)
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(Smeh)
00:22
Well, the topic I'm going to discuss
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Znači, tema o kojoj ću razgovarati
00:24
is one which is, in a certain sense, very peculiar
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je u izvesnom smislu dosta neobična
00:27
because it's very old.
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jer je vrlo stara.
00:29
Roughness is part of human life
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Hrapavost je deo ljudskog života,
00:32
forever and forever,
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i tako će biti za sva vremena.
00:34
and ancient authors have written about it.
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Još su drevni pisci pisali o tome.
00:37
It was very much uncontrollable,
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Nju je uglavnom nemoguće kontrolisati.
00:39
and in a certain sense,
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I u izvesnom smislu,
00:41
it seemed to be the extreme of complexity,
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čini se da je krajnje složena,
00:44
just a mess, a mess and a mess.
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samo zbrka i nered.
00:46
There are many different kinds of mess.
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Postoji mnogo različitih vidova nereda.
00:48
Now, in fact,
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Zapravo,
00:50
by a complete fluke,
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zahvaljujući potpuno srećnom slučaju,
00:52
I got involved many years ago
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pre nekoliko godina sam se uključio
00:55
in a study of this form of complexity,
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u izučavanje ovog vida kompleksnosti.
00:58
and to my utter amazement,
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Na moje kranje čuđenje,
01:00
I found traces --
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pronašao sam tragove –
01:02
very strong traces, I must say --
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veoma snažne tragove, moram reći –
01:04
of order in that roughness.
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reda u toj hrapavosti.
01:07
And so today, I would like to present to you
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Stoga ću danas da vam predstavim
01:09
a few examples
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nekoliko primera
01:11
of what this represents.
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onoga što to predstavlja.
01:13
I prefer the word roughness
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Više mi se sviđa reč hrapavost
01:15
to the word irregularity
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od nepravilnosti,
01:17
because irregularity --
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jer nepravilnost –
01:19
to someone who had Latin
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za nekoga ko je učio latinski
01:21
in my long-past youth --
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u dalekoj mladosti –
01:23
means the contrary of regularity.
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znači nešto što je suprotno pravilnosti.
01:25
But it is not so.
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Ali nije.
01:27
Regularity is the contrary of roughness
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Pravilnost je suprotnost hrapavosti
01:30
because the basic aspect of the world
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jer je osnovni aspekt sveta
01:32
is very rough.
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veoma hrapav.
01:34
So let me show you a few objects.
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Dopustite mi da vam pokažem nekoliko predmeta.
01:37
Some of them are artificial.
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Neki od njih su veštački.
01:39
Others of them are very real, in a certain sense.
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Neki su stvarni, u izvesnom smislu.
01:42
Now this is the real. It's a cauliflower.
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Recimo, ovaj je pravi. Ovo je karfiol.
01:45
Now why do I show a cauliflower,
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Ali zašto pokazujem karfiol,
01:48
a very ordinary and ancient vegetable?
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jedno jako obično i drevno povrće?
01:51
Because old and ancient as it may be,
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Zbog toga što, koliko god da je staro i drevno,
01:54
it's very complicated and it's very simple,
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ono je veoma komplikovano i veoma jednostavno
01:57
both at the same time.
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u isto vreme.
01:59
If you try to weigh it -- of course it's very easy to weigh it,
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Ako pokušate da mu izmerite težinu, naravno, to je jako jednostavno.
02:02
and when you eat it, the weight matters --
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A kada ga jedete, težina je bitna.
02:05
but suppose you try to
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No, pretpostavite da pokušate
02:08
measure its surface.
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da izmerite njegovu površinu.
02:10
Well, it's very interesting.
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Da, to je jako zanimljivo.
02:12
If you cut, with a sharp knife,
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Ako oštrim nožem odsečete
02:15
one of the florets of a cauliflower
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jedan od njegovih cvetova
02:17
and look at it separately,
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i pogledate ga odvojeno,
02:19
you think of a whole cauliflower, but smaller.
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pomislićete na celi karfiol, samo manji.
02:22
And then you cut again,
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I onda nastavite da režete,
02:24
again, again, again, again, again, again, again, again,
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dalje, dalje, dalje, dalje, dalje, dalje, dalje, dalje.
02:27
and you still get small cauliflowers.
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I stalno dobijate male karfiole.
02:29
So the experience of humanity
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Iskustvo čovečanstva
02:31
has always been that there are some shapes
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pokazuje da oduvek postoje oblici
02:34
which have this peculiar property,
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koji poseduju ovakav poseban kvalitet,
02:36
that each part is like the whole,
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gde svaki deo izgleda kao celina,
02:39
but smaller.
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samo manja.
02:41
Now, what did humanity do with that?
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A šta je čovečanstvo učinilo sa time?
02:44
Very, very little.
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Veoma, veoma malo.
02:47
(Laughter)
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(Smeh)
02:50
So what I did actually is to
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Ono što sam ja zapravo učinio
02:53
study this problem,
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je da sam proučavao ovaj problem
02:56
and I found something quite surprising.
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i otkrio nešto sasvim iznenađujuće.
02:59
That one can measure roughness
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Otkrio sam da je moguće izmeriti hrapavost
03:02
by a number, a number,
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nekim brojem, brojem,
03:05
2.3, 1.2 and sometimes much more.
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2.3, 1.2, a ponekad i nekim većim.
03:08
One day, a friend of mine,
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Jedan dan, moj prijatelj je,
03:10
to bug me,
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kako bi me zavitlavao,
03:12
brought a picture and said,
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doneo sliku i rekao
03:14
"What is the roughness of this curve?"
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„Kolika je hrapavost ove obline?“
03:16
I said, "Well, just short of 1.5."
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Odgovorio sam, „Pa, nešto manja od 1.5.“
03:19
It was 1.48.
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Bila je 1.48.
03:21
Now, it didn't take me any time.
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Znači, uopšte nije bilo potrebno mnogo vremena za to,
03:23
I've been looking at these things for so long.
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jer sam toliko dugo posmatrao te stvari.
03:25
So these numbers are the numbers
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Znači, ovi brojevi su brojevi
03:27
which denote the roughness of these surfaces.
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koji označavaju hrapavost ovih površina.
03:30
I hasten to say that these surfaces
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Želim da što pre kažem da su ove površine
03:32
are completely artificial.
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potpuno veštačke.
03:34
They were done on a computer,
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Napravljene su kompjuterski.
03:36
and the only input is a number,
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I jedini unos je broj.
03:38
and that number is roughness.
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A taj broj je hrapavost.
03:41
So on the left,
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I tako sam na levoj strani
03:43
I took the roughness copied from many landscapes.
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uzeo hrapavost preuzetu iz mnogih pejzaža.
03:46
To the right, I took a higher roughness.
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Na desnoj strani, uzeo sam malo više hrapavosti.
03:49
So the eye, after a while,
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I tako oko, nakon nekog vremena,
03:51
can distinguish these two very well.
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može jako dobro napraviti razliku između jednoga i drugoga.
03:54
Humanity had to learn about measuring roughness.
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Čovečanstvo mora da nauči o merenju hrapavosti.
03:56
This is very rough, and this is sort of smooth, and this perfectly smooth.
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Ovo je hrapavo, ovo je donekle glatko, a ovo je savršeno glatko
03:59
Very few things are very smooth.
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Izuzetno malo toga je izuzetno glatko.
04:03
So then if you try to ask questions:
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Pokušajte onda da upitate:
04:06
"What's the surface of a cauliflower?"
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kolika je površina karfiola?
04:08
Well, you measure and measure and measure.
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Možete meriti i meriti i meriti.
04:11
Each time you're closer, it gets bigger,
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Svaki put kada se približite on postane još veći,
04:14
down to very, very small distances.
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čak i do najmanjih razmaka.
04:16
What's the length of the coastline
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Kolika je dužina obale
04:18
of these lakes?
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ovih jezera?
04:20
The closer you measure, the longer it is.
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Što bliže merite, biće sve duža.
04:23
The concept of length of coastline,
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Koncept dužine obale,
04:25
which seems to be so natural
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koja se čini tako prirodnom
04:27
because it's given in many cases,
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jer se u mnogim slučajevima uzima zdravo za gotovo,
04:29
is, in fact, complete fallacy; there's no such thing.
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zapravo je potpuna obmana; ne postoji ništa takvo.
04:32
You must do it differently.
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Morate da pokušate na drugačiji način.
04:35
What good is that, to know these things?
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Kakva je korist znati sve ove stvari?
04:37
Well, surprisingly enough,
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Hm, dovoljno iznenađujuće,
04:39
it's good in many ways.
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korisno je na mnogo načina.
04:41
To begin with, artificial landscapes,
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Za početak, veštački pejzaži
04:43
which I invented sort of,
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koje sam na neki način izmislio,
04:45
are used in cinema all the time.
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koriste se u kinematografiji sve vreme.
04:48
We see mountains in the distance.
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Vidimo planine u daljini.
04:50
They may be mountains, but they may be just formulae, just cranked on.
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Mogle bi biti planine, ali bi mogle biti i formule, samo sklepane.
04:53
Now it's very easy to do.
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U današnje vreme je to jako jednostavno učiniti.
04:55
It used to be very time-consuming, but now it's nothing.
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Nekada je za to bilo potrebno mnogo vremena, ali sada je to sitnica.
04:58
Now look at that. That's a real lung.
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Pogledajte. Ovo su prava pluća.
05:01
Now a lung is something very strange.
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Pluća su nešto jako čudno.
05:03
If you take this thing,
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Ako ih uzmete,
05:05
you know very well it weighs very little.
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znate jako dobro da je njihova težina mala.
05:08
The volume of a lung is very small,
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Zapremina pluća je veoma mala.
05:10
but what about the area of the lung?
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Ali kolika je površina pluća?
05:13
Anatomists were arguing very much about that.
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Anatomi su se mnogo svađali oko toga.
05:16
Some say that a normal male's lung
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Neki kažu da pluća normalnog muškarca
05:19
has an area of the inside
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imaju površinu unutrašnjosti
05:21
of a basketball [court].
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košarkaškog igrališta.
05:23
And the others say, no, five basketball [courts].
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A oni drugi kažu – ne, pet košarkaških igrališta.
05:27
Enormous disagreements.
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Ogromna razlika u mišljenju.
05:29
Why so? Because, in fact, the area of the lung
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Zbog čega? Zato što je, zapravo, površina pluća
05:32
is something very ill-defined.
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nešto što je jako loše definisano.
05:35
The bronchi branch, branch, branch
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Bronhije se razgranavaju, razgranavaju, razgranavaju.
05:38
and they stop branching,
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I onda prestanu da se granaju,
05:41
not because of any matter of principle,
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ne zbog bilo kog posebnog principa,
05:44
but because of physical considerations:
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već zbog fizičkih okolnosti,
05:47
the mucus, which is in the lung.
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sluzi koja se nalazi u plućima.
05:50
So what happens is that in a way
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Znači, ono što se dešava jeste da na taj način
05:52
you have a much bigger lung,
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imate mnogo veća pluća,
05:54
but it branches and branches
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ali ako se razgranavaju i razgranavaju,
05:56
down to distances about the same for a whale, for a man
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sve do rastojanja koja su gotovo jednaka kod kita, čoveka
05:59
and for a little rodent.
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ili malog glodara.
06:02
Now, what good is it to have that?
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Ali, kakva je korist od toga?
06:05
Well, surprisingly enough, amazingly enough,
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Pa, dovoljno iznenađujuće, dovoljno začuđujuće,
06:07
the anatomists had a very poor idea
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je da su anatomi imali jako bledu ideju
06:10
of the structure of the lung until very recently.
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o strukturi pluća sve donedavno.
06:13
And I think that my mathematics,
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A ja mislim da je moja matematika,
06:15
surprisingly enough,
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sasvim začuđujuće,
06:17
has been of great help
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bila od velike pomoći
06:19
to the surgeons
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hirurzima
06:21
studying lung illnesses
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koji izučavaju bolesti pluća
06:23
and also kidney illnesses,
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kao i bolesti bubrega,
06:25
all these branching systems,
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sve te sisteme koji se razgranjavaju,
06:27
for which there was no geometry.
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za koje nije postojala nikakva geometrija.
06:30
So I found myself, in other words,
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I tako sam se našao kako, drugim rečima,
06:32
constructing a geometry,
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konstruišem geometriju,
06:34
a geometry of things which had no geometry.
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geometriju stvari koje nemaju geometriju.
06:37
And a surprising aspect of it
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A iznenađujući aspekt toga je
06:39
is that very often, the rules of this geometry
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da su veoma često pravila ove geometrije
06:42
are extremely short.
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izuzetno kratka.
06:44
You have formulas that long.
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Imate formule toliko dugačke.
06:46
And you crank it several times.
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I onda ih izvršite nekoliko puta.
06:48
Sometimes repeatedly: again, again, again,
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Ponekad to ponovite više puta.
06:50
the same repetition.
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Isto ponavljanje.
06:52
And at the end, you get things like that.
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I na kraju dobijete nešto poput toga.
06:54
This cloud is completely,
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Ovaj oblak je potpuno,
06:56
100 percent artificial.
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stoprocentno veštački.
06:59
Well, 99.9.
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Pa dobro, 99.9.
07:01
And the only part which is natural
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A jedini deo koji je prirodan
07:03
is a number, the roughness of the cloud,
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je broj, hrapavost oblaka,
07:05
which is taken from nature.
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koji je preuzet iz prirode.
07:07
Something so complicated like a cloud,
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Iza nečega tako komplikovanog poput oblaka,
07:09
so unstable, so varying,
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tako nestalnog, tako promenljivog,
07:11
should have a simple rule behind it.
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bi trebalo postojati jednostavno pravilo.
07:14
Now this simple rule
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Ali to jednostavno pravilo
07:17
is not an explanation of clouds.
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nije objašnjenje za oblake.
07:20
The seer of clouds had to
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Posmatrač oblaka mora
07:22
take account of it.
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to uzeti u obzir.
07:24
I don't know how much advanced
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Nisam siguran koliko su unapređene
07:27
these pictures are. They're old.
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ove slike, jer su stare.
07:29
I was very much involved in it,
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Mnogo sam se bavio njima, .
07:31
but then turned my attention to other phenomena.
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ali onda sam preusmerio pažnju ne druge fenomene.
07:34
Now, here is another thing
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Ovde imamo još jednu stvar
07:36
which is rather interesting.
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koja je dosta zanimljiva.
07:39
One of the shattering events
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Jedan od dramatičnih događaja
07:41
in the history of mathematics,
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u istoriji matematike,
07:43
which is not appreciated by many people,
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a koji veliki broj ljudi uopšte ne ceni,
07:46
occurred about 130 years ago,
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desio se pre nekih 130,
07:48
145 years ago.
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145 godina.
07:50
Mathematicians began to create
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Matematičari su počeli da stvaraju
07:52
shapes that didn't exist.
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oblike koji nisu postojali.
07:54
Mathematicians got into self-praise
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Matematičari su počeli sa hvalospevima samim sebi
07:57
to an extent which was absolutely amazing,
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do te mere gde je potpuno neverovatno
07:59
that man can invent things
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da čovek može izumeti stvari
08:01
that nature did not know.
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koje sama priroda nije poznavala.
08:03
In particular, it could invent
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A posebno da može izumeti
08:05
things like a curve which fills the plane.
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stvari poput krive koja ispunjava ravan
08:08
A curve's a curve, a plane's a plane,
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Kriva je kriva, ravan je ravan,
08:10
and the two won't mix.
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i jedno sa drugim se ne meša.
08:12
Well, they do mix.
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Ali, ipak se mešaju.
08:14
A man named Peano
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Čovek po imenu Peano
08:16
did define such curves,
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je definisao takve krive,
08:18
and it became an object of extraordinary interest.
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koje su postale predmet neverovatnog interesovanja.
08:21
It was very important, but mostly interesting
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To je bilo jako bitno, ali uglavnom zanimljivo
08:24
because a kind of break,
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zbog izvesnog prekida,
08:26
a separation between
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razdvajanja između
08:28
the mathematics coming from reality, on the one hand,
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matematike koja proizilazi iz realnosti, s jedne strane
08:31
and new mathematics coming from pure man's mind.
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i nove matematike koja dolazi iz čistog uma čoveka.
08:34
Well, I was very sorry to point out
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Bilo mi je jako žao što sam morao ukazati
08:37
that the pure man's mind
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da je čisti um čoveka
08:39
has, in fact,
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zapravo
08:41
seen at long last
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konačno video
08:43
what had been seen for a long time.
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ono što se moglo videti već dugo vremena.
08:45
And so here I introduce something,
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I tako sam ovde uveo nešto novo,
08:47
the set of rivers of a plane-filling curve.
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niz reka krive koja ispunjava ravan.
08:50
And well,
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Ali,
08:52
it's a story unto itself.
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to je priča sama za sebe.
08:54
So it was in 1875 to 1925,
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Dakle, između 1875. i 1925.,
08:57
an extraordinary period
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u to neverovatno doba
08:59
in which mathematics prepared itself to break out from the world.
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matematika se pripremala da se odvoji od sveta.
09:02
And the objects which were used
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A predmete koji su korišćeni
09:04
as examples, when I was
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kao primeri, u vreme kada sam ja bio
09:06
a child and a student, as examples
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dete i student,
09:08
of the break between mathematics
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primeri prekida između matematike
09:11
and visible reality --
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i vidljive realnosti –
09:13
those objects,
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te predmete
09:15
I turned them completely around.
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sam potpuno preokrenuo.
09:17
I used them for describing
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Koristio sam ih kako bih opisao
09:19
some of the aspects of the complexity of nature.
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neke od aspekata složenosti prirode.
09:22
Well, a man named Hausdorff in 1919
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Čovek po imenu Hausdorf je 1919. godine
09:25
introduced a number which was just a mathematical joke,
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uveo broj koji je bio samo matematička šala.
09:28
and I found that this number
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A ja sam otkrio da je taj broj
09:30
was a good measurement of roughness.
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dobra mera hrapavosti.
09:32
When I first told it to my friends in mathematics
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Kada sam ga prvi put saopštio mojim prijateljima matematičarima
09:34
they said, "Don't be silly. It's just something [silly]."
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rekli su, "Ne budi lud. To je tek nešto (blesavo)".
09:37
Well actually, I was not silly.
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No, zapravo, uopšte nije bilo blesavo.
09:40
The great painter Hokusai knew it very well.
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Čuveni slikar Hokusai je to jako dobro znao.
09:43
The things on the ground are algae.
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Ovo su alge.
09:45
He did not know the mathematics; it didn't yet exist.
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On nije bio upoznat sa matematikom; u to vreme još nije postojala.
09:48
And he was Japanese who had no contact with the West.
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A on je bio Japanac bez ikakvog kontakta sa zapadnim svetom.
09:51
But painting for a long time had a fractal side.
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No, njegove su slike odavno imale odliku fraktala.
09:54
I could speak of that for a long time.
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Mogao bih nadugo govoriti o ovome.
09:56
The Eiffel Tower has a fractal aspect.
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Ajfelov toranj ima fraktalni aspekt.
09:59
I read the book that Mr. Eiffel wrote about his tower,
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A pročitao sam i knjigu koju je g-din Ajfel napisao o svom tornju.
10:02
and indeed it was astonishing how much he understood.
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I, zaista, bilo je zapanjujuće koliko mnogo je on razumeo.
10:05
This is a mess, mess, mess, Brownian loop.
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To je zbrka, zbrka, zbrka, Braunovo kretanje.
10:08
One day I decided --
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Jednog dana,
10:10
halfway through my career,
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na pola svoja karijere,
10:12
I was held by so many things in my work --
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toliko toga me je usporavalo u mom poslu,
10:15
I decided to test myself.
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i odlučio sam da se podvrgnem testu.
10:18
Could I just look at something
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Da li bih mogao gledati u nešto
10:20
which everybody had been looking at for a long time
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što svi posmatraju već jako dugo vremena
10:23
and find something dramatically new?
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i pronaći nešto dramatično novo?
10:26
Well, so I looked at these
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I tako sam gledao u ove
10:29
things called Brownian motion -- just goes around.
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stvari koje se nazivaju Braunovim kretanjem – i tako u krug.
10:32
I played with it for a while,
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Neko vreme sam se igrao sa tim
10:34
and I made it return to the origin.
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i vratio je na početak.
10:37
Then I was telling my assistant,
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Govorio sam svome asistentu
10:39
"I don't see anything. Can you paint it?"
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„Ništa ne vidim. Možeš li ti to naslikati?“
10:41
So he painted it, which means
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I on je naslikao, što znači
10:43
he put inside everything. He said:
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da je sve stavio unutra. Rekao je:
10:45
"Well, this thing came out ..." And I said, "Stop! Stop! Stop!
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„Hm, ovo je izašlo....“ A ja sam rekao „Stani! Stani! Stani!
10:48
I see; it's an island."
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Vidim ga, to je ostrvo."
10:51
And amazing.
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Zapanjujuće.
10:53
So Brownian motion, which happens to have
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Znači, Braunovo kretanje, koja ima
10:55
a roughness number of two, goes around.
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hrapavost broja dva, se okreće.
10:58
I measured it, 1.33.
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Izmerio sam ga, 1.33.
11:00
Again, again, again.
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Opet, i opet, i opet.
11:02
Long measurements, big Brownian motions,
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Duga merenja, velika Braunova kretanja,
11:04
1.33.
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1.33.
11:06
Mathematical problem: how to prove it?
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Matematički problem: kako ga dokazati?
11:09
It took my friends 20 years.
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Mojim prijateljima je trebalo 20 godina.
11:12
Three of them were having incomplete proofs.
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Trojica njih su imali nepotpune dokaze.
11:15
They got together, and together they had the proof.
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Oni su se sastali i tek zajedno su imali dokaz.
11:19
So they got the big [Fields] medal in mathematics,
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Tako su dobili veliku medalju u matematici,
11:22
one of the three medals that people have received
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jednu od tri priznanja koja pojedinci primaju
11:24
for proving things which I've seen
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za dokazivanje stvari koje sam video
11:27
without being able to prove them.
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ali nisam bio u stanju da dokažem.
11:30
Now everybody asks me at one point or another,
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I sada me svi pre ili kasnije pitaju
11:33
"How did it all start?
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„Kako je to sve počelo?
11:35
What got you in that strange business?"
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Šta te je navelo na taj čudni posao?“
11:38
What got you to be,
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Šta me je navelo
11:40
at the same time, a mechanical engineer,
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da istovremeno budem mehanički inženjer,
11:42
a geographer
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geograf
11:44
and a mathematician and so on, a physicist?
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i matematičar, i tako dalje, fizičar?
11:46
Well actually I started, oddly enough,
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Hm, zapravo sam počeo
11:49
studying stock market prices.
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izučavati cene na berzi.
11:51
And so here
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I tako sam
11:53
I had this theory,
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imao ovu teoriju,
11:56
and I wrote books about it --
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i pisao sam knjige o tome.
11:58
financial prices increments.
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Rast finansijskih cena.
12:00
To the left you see data over a long period.
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Na levoj strani vidite podatke prikupljane tokom dugo vremena.
12:02
To the right, on top,
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Na desnoj, na vrhu,
12:04
you see a theory which is very, very fashionable.
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vidite teoriju koja je veoma, veoma u modi.
12:07
It was very easy, and you can write many books very fast about it.
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Bilo je jako jednostavno i jako brzo možete napisati mnogo knjiga o tome.
12:10
(Laughter)
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(Smeh)
12:12
There are thousands of books on that.
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Postoji na hiljade knjiga o tome.
12:15
Now compare that with real price increments.
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Sada uporedite to sa stvarnim uvećanjem cena
12:18
Where are real price increments?
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i gde su prava uvećanja cena?
12:20
Well, these other lines
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Ove druge linije
12:22
include some real price increments
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uključuju neka realna uvećanja cena,
12:24
and some forgery which I did.
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i neka lažna, koja sam ja uneo.
12:26
So the idea there was
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Znači, ideja je bila
12:28
that one must be able to -- how do you say? --
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da je moguće – kako biste to rekli? –
12:30
model price variation.
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napraviti model varijacije cena.
12:33
And it went really well 50 years ago.
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I to je išlo jako dobro pre 50 godina.
12:36
For 50 years, people were sort of pooh-poohing me
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Pedeset godina ljudi su me na neki način omalovažavali
12:39
because they could do it much, much easier.
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jer su oni to mogli uraditi na mnogo, mnogo jednostavniji način.
12:41
But I tell you, at this point, people listened to me.
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Ali da vam kažem, ovoga puta, ljudi su me slušali.
12:44
(Laughter)
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(Smeh)
12:46
These two curves are averages:
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Ove dve krive su prosečne vrednosti.
12:48
Standard & Poor, the blue one;
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S&P (agencija za kreditne rejtinge), plave boje.
12:50
and the red one is Standard & Poor's
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A crvena je S&P,
12:52
from which the five biggest discontinuities
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iz koje je pet najvećih diskontinuiteta
12:55
are taken out.
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izdvojeno.
12:57
Now discontinuities are a nuisance,
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No, diskontinuiteti su smetnja.
12:59
so in many studies of prices,
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U mnogim studijama cena,
13:02
one puts them aside.
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oni se ostavljaju po strani.
13:04
"Well, acts of God.
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"Pa, više sile.
13:06
And you have the little nonsense which is left.
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I imate malo besmislenost koje je ostalo.
13:09
Acts of God." In this picture,
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Više sile." Na ovoj slici
13:12
five acts of God are as important as everything else.
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pet takvih sila su podjednako važne kao i sve ostalo.
13:15
In other words,
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Drugim rečima,
13:17
it is not acts of God that we should put aside.
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nije viša sila to što bismo trebali staviti po strani.
13:19
That is the meat, the problem.
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To je suština, problem.
13:22
If you master these, you master price,
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Ako ovladate njima, ovladali ste cenom.
13:25
and if you don't master these, you can master
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Ako ne ovladate njima, možete ovladati
13:27
the little noise as well as you can,
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nad ovo malo šuma koliko god želite,
13:29
but it's not important.
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no to nije važno.
13:31
Well, here are the curves for it.
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Evo, ovde su krive koje to pokazuju.
13:33
Now, I get to the final thing, which is the set
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Sada ću preći na poslednju stvar, a to je niz
13:35
of which my name is attached.
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od kog je sačinjeno moje ime.
13:37
In a way, it's the story of my life.
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Na neki način, to je priča mog života.
13:39
My adolescence was spent
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Svoje doba adolescencije sam proveo
13:41
during the German occupation of France.
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tokom nemačke okupacije Francuske.
13:43
Since I thought that I might
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I s obzirom da sam mislio kako bih
13:46
vanish within a day or a week,
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mogao nestati kroz dan ili nedelju,
13:49
I had very big dreams.
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imao sam velike snove.
13:52
And after the war,
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I nakon rata
13:54
I saw an uncle again.
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sam ponovo video svog strica.
13:56
My uncle was a very prominent mathematician, and he told me,
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Moj stric je bio jako istaknut matematičar i rekao mi je
13:58
"Look, there's a problem
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„Pazi, ima jedan problem
14:00
which I could not solve 25 years ago,
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koji nisam mogao rešiti pre 25 godina,
14:02
and which nobody can solve.
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i koji niko ne može da reši.
14:04
This is a construction of a man named [Gaston] Julia
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To je konstrukcija čoveka po imenu (Gaston) Julia
14:06
and [Pierre] Fatou.
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i (Pierre) Fatou.
14:08
If you could
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Ako bi uspeo
14:10
find something new, anything,
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da nađeš nešto novo, bilo šta,
14:12
you will get your career made."
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napravićeš karijeru“.
14:14
Very simple.
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Veoma jednostavno.
14:16
So I looked,
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I tako sam počeo tražiti,
14:18
and like the thousands of people that had tried before,
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i kao i hiljade onih koji su pokušali pre mene,
14:20
I found nothing.
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nisam ništa našao.
14:23
But then the computer came,
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A onda se pojavio kompjuter.
14:25
and I decided to apply the computer,
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I ja sam odlučio da primenim kompjuter,
14:27
not to new problems in mathematics --
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ne na nove matematičke probleme –
14:30
like this wiggle wiggle, that's a new problem --
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poput ovog mrdanja ovde, to je novi problem –
14:32
but to old problems.
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već na stare probleme.
14:34
And I went from what's called
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I krenuo sam od onoga što se zove
14:36
real numbers, which are points on a line,
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realni brojevi, koji predstavljaju tačke na liniji,
14:38
to imaginary, complex numbers,
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ka imaginarnim, složenim brojevima,
14:40
which are points on a plane,
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koji su tačke na ravni,
14:42
which is what one should do there,
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i to je ono što treba da se tu uradi.
14:44
and this shape came out.
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I dobio sam ovaj oblik.
14:46
This shape is of an extraordinary complication.
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Ovaj oblik je neverovatno komplikovan.
14:49
The equation is hidden there,
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Ovde je skrivena jednačina,
14:51
z goes into z squared, plus c.
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z ide u z na kvadrat, plus c.
14:54
It's so simple, so dry.
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Jako je jednostavno, jako suvo.
14:56
It's so uninteresting.
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I tako nezanimljivo.
14:58
Now you turn the crank once, twice:
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Okrenite ovaj kotur jednom, drugi put,
15:01
twice,
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još jednom,
15:04
marvels come out.
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i izaći će čudo.
15:06
I mean this comes out.
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Mislim, ovo će izaći.
15:08
I don't want to explain these things.
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Ne želim da objašnjavam ove stvari.
15:10
This comes out. This comes out.
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Pojavljuje se ovo. I ovo.
15:12
Shapes which are of such complication,
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Oblici koji su tako komplikovani,
15:14
such harmony and such beauty.
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takvog sklada i lepote.
15:17
This comes out
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Ovo izlazi
15:19
repeatedly, again, again, again.
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ponovo, iznova i iznova.
15:21
And that was one of my major discoveries,
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I to je bilo jedno od mojih glavnih otkrića
15:23
to find that these islands were the same
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da su ova ostrva ista
15:25
as the whole big thing, more or less.
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kao i cela velika stvar, manje ili više.
15:27
And then you get these
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A onda imate ove
15:29
extraordinary baroque decorations all over the place.
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neobične barokne dekoracije posvuda.
15:32
All that from this little formula,
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I sve to iz ove male formule,
15:35
which has whatever, five symbols in it.
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koja se sastoji od koliko god, nekih pet simbola.
15:38
And then this one.
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A potom ova.
15:40
The color was added for two reasons.
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Boja je dodana iz dva razloga.
15:42
First of all, because these shapes
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Najpre zbog toga što su ovi oblici
15:44
are so complicated
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previše komplikovani
15:47
that one couldn't make any sense of the numbers.
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da bi iko mogao videti bilo kakav smisao u brojevima.
15:50
And if you plot them, you must choose some system.
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A ako ih unosite u grafikon, morate odabrati neki sistem.
15:53
And so my principle has been
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A moj princip je oduvek bio
15:55
to always present the shapes
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da predstavim oblike
15:58
with different colorings
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različitim bojama,
16:00
because some colorings emphasize that,
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zato što neke boje naglašavaju ovo,
16:02
and others it is that or that.
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a neke druge ovo ili ono.
16:04
It's so complicated.
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Tako je komplikovano.
16:06
(Laughter)
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(Smeh)
16:08
In 1990, I was in Cambridge, U.K.
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1990. godine sam bio u Kembridžu, u Ujedinjenom Kraljevstvu
16:10
to receive a prize from the university,
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kako bih primio nagradu univerziteta.
16:13
and three days later,
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I tri dana nakon toga,
16:15
a pilot was flying over the landscape and found this thing.
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jedan pilot je leteo iznad pejzaža i otkrio ovo.
16:18
So where did this come from?
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Odakle se ovo pojavilo?
16:20
Obviously, from extraterrestrials.
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Očigledno, od vanzemaljaca.
16:22
(Laughter)
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(Smeh)
16:25
Well, so the newspaper in Cambridge
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Tako su novine u Kembridžu
16:27
published an article about that "discovery"
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objavile članak o „otkriću“
16:29
and received the next day
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i sledeći dan primile
16:31
5,000 letters from people saying,
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5.000 pisama od čitalaca koji su rekli
16:33
"But that's simply a Mandelbrot set very big."
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„Ali to je naprosto veoma veliki Mandelbrotov set“.
16:37
Well, let me finish.
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Dopustite mi da završim.
16:39
This shape here just came
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Ovaj oblik ovde je nastao
16:41
out of an exercise in pure mathematics.
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kao vežba u čistoj matematici.
16:43
Bottomless wonders spring from simple rules,
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Beskrajna čuda izviru iz jednostavnih pravila,
16:46
which are repeated without end.
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koja se beskrajno ponavljaju.
16:49
Thank you very much.
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Hvala vam mnogo.
16:51
(Applause)
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Aplauz.
ABOUT THE SPEAKER
Benoit Mandelbrot - MathematicianBenoit Mandelbrot's work led the world to a deeper understanding of fractals, a broad and powerful tool in the study of roughness, both in nature and in humanity's works.
Why you should listen
Studying complex dynamics in the 1970s, Benoit Mandelbrot had a key insight about a particular set of mathematical objects: that these self-similar structures with infinitely repeating complexities were not just curiosities, as they'd been considered since the turn of the century, but were in fact a key to explaining non-smooth objects and complex data sets -- which make up, let's face it, quite a lot of the world. Mandelbrot coined the term "fractal" to describe these objects, and set about sharing his insight with the world.
The Mandelbrot set (expressed as z² + c) was named in Mandelbrot's honor by Adrien Douady and John H. Hubbard. Its boundary can be magnified infinitely and yet remain magnificently complicated, and its elegant shape made it a poster child for the popular understanding of fractals. Led by Mandelbrot's enthusiastic work, fractal math has brought new insight to the study of pretty much everything, from the behavior of stocks to the distribution of stars in the universe.
Benoit Mandelbrot appeared at the first TED in 1984, and returned in 2010 to give an overview of the study of fractals and the paradigm-flipping insights they've brought to many fields. He died in October 2010 at age 85. Read more about his life on NYBooks.com >>
Benoit Mandelbrot | Speaker | TED.com