TED2010
Benoit Mandelbrot: Fractals and the art of roughness
Benoit Mandelbrot: Fraktály a umenie drsnosti
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Na TED2010, matematická legenda Benoit Mandelbrot rozvíja tému, o ktorej prvýkrát hovoril na TEDe v roku 1984 -- extrémna komplexnosť drsnosti a spôsob, ako vie fraktálna matematika objaviť poriadok vo vzoroch, ktoré vyzerajú nesmierne komplikované.
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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Ďakujem veľmi pekne.
00:17
Please excuse me for sitting; I'm very old.
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Prosím ospravedlňte, že sedím: Som veľmi starý.
00:20
(Laughter)
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(Smiech)
00:22
Well, the topic I'm going to discuss
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Nuž, téma o ktorej budem rozprávať,
00:24
is one which is, in a certain sense, very peculiar
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je v istom zmysle veľmi špecifická,
00:27
because it's very old.
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pretože je veľmi stará.
00:29
Roughness is part of human life
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Drsnosť je súčasť ľudského života
00:32
forever and forever,
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od nepamäti.
00:34
and ancient authors have written about it.
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Už dávni autori o nej písali.
00:37
It was very much uncontrollable,
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Je veľmi nekontrolovateľná.
00:39
and in a certain sense,
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A v istom zmysle
00:41
it seemed to be the extreme of complexity,
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sa zdalo, že je to extrém zložitosti,
00:44
just a mess, a mess and a mess.
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proste neporiadok, neporiadok, neporiadok.
00:46
There are many different kinds of mess.
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Sú rôzne druhy neporiadku.
00:48
Now, in fact,
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Takže,
00:50
by a complete fluke,
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úplnou náhodou
00:52
I got involved many years ago
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som sa dostal veľmi dávno
00:55
in a study of this form of complexity,
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k štúdiu tejto formy zložitosti.
00:58
and to my utter amazement,
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A na moje hlboké počudovanie,
01:00
I found traces --
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som našiel stopy --
01:02
very strong traces, I must say --
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veľmi silné stopy, musím povedať --
01:04
of order in that roughness.
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poriadku v tej drsnosti.
01:07
And so today, I would like to present to you
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A dnes by som vám rád prezentoval
01:09
a few examples
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zopár príkladov toho,
01:11
of what this represents.
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čo to predstavuje.
01:13
I prefer the word roughness
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Preferujem slovo drsnosť
01:15
to the word irregularity
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pred slovom nepravidelnosť,
01:17
because irregularity --
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pretože nepravidelnosť --
01:19
to someone who had Latin
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pre niekoho, kto mal latinčinu,
01:21
in my long-past youth --
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v mojej dávnej mladosti --
01:23
means the contrary of regularity.
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znamená opak pravidelnosti.
01:25
But it is not so.
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Ale tak to nie je.
01:27
Regularity is the contrary of roughness
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Pravidelnosť je opakom drsnosti,
01:30
because the basic aspect of the world
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pretože základný aspekt sveta
01:32
is very rough.
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je veľmi drsný.
01:34
So let me show you a few objects.
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Ukážem vám pár objektov.
01:37
Some of them are artificial.
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Niektoré z nich sú umelé.
01:39
Others of them are very real, in a certain sense.
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Niektoré sú veľmi reálne, v istom zmysle.
01:42
Now this is the real. It's a cauliflower.
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Toto je ten reálny. Je to karfiol.
01:45
Now why do I show a cauliflower,
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Ale prečo ukazujem karfiol,
01:48
a very ordinary and ancient vegetable?
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veľmi obyčajnú a pradávnu zeleninu?
01:51
Because old and ancient as it may be,
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Pretože tak aký je starý a pradávny,
01:54
it's very complicated and it's very simple,
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je aj veľmi komplikovaný a veľmi jednoduchý
01:57
both at the same time.
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zároveň.
01:59
If you try to weigh it -- of course it's very easy to weigh it,
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Ak ho skúsite odvážiť, samozrejme, je ľahké ho odvážiť.
02:02
and when you eat it, the weight matters --
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A keď ho jete, záleží na váhe.
02:05
but suppose you try to
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Ale čo keď skúsite
02:08
measure its surface.
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zmerať jeho povrch.
02:10
Well, it's very interesting.
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Nuž, to je veľmi zaujímavé.
02:12
If you cut, with a sharp knife,
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Ak odrežete ostrým nožom
02:15
one of the florets of a cauliflower
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jeden kvietok karfiolu
02:17
and look at it separately,
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a pozriete sa naň samostatne,
02:19
you think of a whole cauliflower, but smaller.
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vidíte celý karfiol, len menší.
02:22
And then you cut again,
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A tak ho rozkrojíte znova,
02:24
again, again, again, again, again, again, again, again,
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znova, znova, znova, znova, znova, znova, znova, znova.
02:27
and you still get small cauliflowers.
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A stále dostávate malé karfioly.
02:29
So the experience of humanity
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Zo skúsenosti ľudstva
02:31
has always been that there are some shapes
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vieme, že sú isté tvary,
02:34
which have this peculiar property,
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ktoré majú túto zvláštnu vlastnosť,
02:36
that each part is like the whole,
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že každá časť je ako celok,
02:39
but smaller.
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len menšia.
02:41
Now, what did humanity do with that?
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A čo s tým ľudstvo spravilo?
02:44
Very, very little.
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Veľmi, veľmi málo.
02:47
(Laughter)
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(Smiech)
02:50
So what I did actually is to
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Tak som sa do toho pustil,
02:53
study this problem,
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študoval som tento problém
02:56
and I found something quite surprising.
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a objavil som niečo dosť prekvapujúce.
02:59
That one can measure roughness
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Že sa dá merať drsnosť
03:02
by a number, a number,
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číslom, číslom,
03:05
2.3, 1.2 and sometimes much more.
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2,3, 1,2 a niekedy oveľa viac.
03:08
One day, a friend of mine,
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Jedného dňa, jeden priateľ,
03:10
to bug me,
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aby ma podpichol,
03:12
brought a picture and said,
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priniesol obrázok a pýta sa,
03:14
"What is the roughness of this curve?"
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"Aká je drsnosť tejto krivky?"
03:16
I said, "Well, just short of 1.5."
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Povedal som, "Nuž, tesne pod 1,5"
03:19
It was 1.48.
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Bolo to 1.48.
03:21
Now, it didn't take me any time.
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Nezabralo to žiadny čas.
03:23
I've been looking at these things for so long.
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Už som sa na tieto veci pozeral tak dlho.
03:25
So these numbers are the numbers
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Tieto čísla sú čísla,
03:27
which denote the roughness of these surfaces.
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ktoré označujú drsnosť týchto povrchov.
03:30
I hasten to say that these surfaces
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Rýchlo dodám, že tieto povrchy
03:32
are completely artificial.
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sú úplne umelé.
03:34
They were done on a computer,
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Vytvoril ich počítač.
03:36
and the only input is a number,
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A jediným vstupom je číslo.
03:38
and that number is roughness.
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A to číslo je drsnosť.
03:41
So on the left,
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Naľavo,
03:43
I took the roughness copied from many landscapes.
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som vzal drsnosť prebranú z prírody.
03:46
To the right, I took a higher roughness.
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Vpravo je vyššia drsnosť.
03:49
So the eye, after a while,
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A oko, po nejakej dobe,
03:51
can distinguish these two very well.
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vie tieto dva pomerne dobre rozlíšiť.
03:54
Humanity had to learn about measuring roughness.
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Ľudstvo sa muselo naučiť merať drsnosť.
03:56
This is very rough, and this is sort of smooth, and this perfectly smooth.
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Toto je veľmi drsné, toto je pomerne hladké a toho je úplne hladké.
03:59
Very few things are very smooth.
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Veľmi málo vecí je hladkých.
04:03
So then if you try to ask questions:
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A potom, ak sa opýtate otázky:
04:06
"What's the surface of a cauliflower?"
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aký je povrch karfiolu?
04:08
Well, you measure and measure and measure.
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Nuž, meriate, meriate a meriate.
04:11
Each time you're closer, it gets bigger,
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Vždy, keď sa priblížite, povrch sa zväčšuje,
04:14
down to very, very small distances.
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až k veľmi, veľmi malým vzdialenostiam.
04:16
What's the length of the coastline
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Aká je dĺžka pobrežia
04:18
of these lakes?
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týchto jazier?
04:20
The closer you measure, the longer it is.
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Čím z bližšej vzdialenosti sa pozeráte, tým je dlhšia.
04:23
The concept of length of coastline,
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Koncept dĺžky pobrežia,
04:25
which seems to be so natural
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čo sa zdá také prirodzené,
04:27
because it's given in many cases,
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pretože sa udáva na veľa miestach,
04:29
is, in fact, complete fallacy; there's no such thing.
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je, vlastne, úplná lož; nič také nie je.
04:32
You must do it differently.
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Musíte to robiť inak.
04:35
What good is that, to know these things?
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Na čo je to teda dobré, vedieť tieto veci?
04:37
Well, surprisingly enough,
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Nuž, prekvapujúco,
04:39
it's good in many ways.
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je to užitočné mnohými spôsobmi.
04:41
To begin with, artificial landscapes,
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Na začiatok, umelé krajiny,
04:43
which I invented sort of,
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ktoré som tak trochu vymyslel,
04:45
are used in cinema all the time.
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sa používajú v kine vlastne stále.
04:48
We see mountains in the distance.
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Vidíme hory na horizonte.
04:50
They may be mountains, but they may be just formulae, just cranked on.
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Môžu to byť hory, ale môžu to byť len vzorce, len uletené.
04:53
Now it's very easy to do.
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Je to vlastne veľmi jednoduché.
04:55
It used to be very time-consuming, but now it's nothing.
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Bývalo to veľmi časovo náročné, ale dnes je to nič.
04:58
Now look at that. That's a real lung.
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Pozrite sa na to. To sú skutočné pľúca.
05:01
Now a lung is something very strange.
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Pľúca sú niečo veľmi čudné.
05:03
If you take this thing,
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Ak vezmete túto vec,
05:05
you know very well it weighs very little.
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veľmi dobre viete, koľko váži.
05:08
The volume of a lung is very small,
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Objem pľúc je veľmi malý.
05:10
but what about the area of the lung?
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Ale ako je to s povrchom pľúc?
05:13
Anatomists were arguing very much about that.
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Anatómovia sa o tom dosť sporili.
05:16
Some say that a normal male's lung
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Niektorí vravia, že normálne mužské pľúca
05:19
has an area of the inside
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majú povrch
05:21
of a basketball [court].
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basketbalového ihriska.
05:23
And the others say, no, five basketball [courts].
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A iní vravia, nie, piatich ihrísk.
05:27
Enormous disagreements.
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Úžasné nezrovnalosti.
05:29
Why so? Because, in fact, the area of the lung
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Prečo to? Lebo, v skutočnosti, povrch pľúc,
05:32
is something very ill-defined.
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je niečo veľmi zle definované.
05:35
The bronchi branch, branch, branch
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Priedušky sa vetvia, vetvia, vetvia.
05:38
and they stop branching,
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A potom sa prestanú vetviť,
05:41
not because of any matter of principle,
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nie kvôli nejakému princípu,
05:44
but because of physical considerations:
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ale kvôli fyzikálnym okolnostiam,
05:47
the mucus, which is in the lung.
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kvôli hlienu, ktorý je v pľúcach.
05:50
So what happens is that in a way
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Takýmto spôsobom
05:52
you have a much bigger lung,
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máte oveľa väčšie pľúca,
05:54
but it branches and branches
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ktoré sa vetvia a vetvia,
05:56
down to distances about the same for a whale, for a man
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až k vzdialenostiam, ktoré sú zhruba rovnaké pre veľryby, pre ľudí
05:59
and for a little rodent.
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a pre malé hlodavce.
06:02
Now, what good is it to have that?
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Na čo je to všetko dobré?
06:05
Well, surprisingly enough, amazingly enough,
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Nuž, prekvapujúco, až udivujúco,
06:07
the anatomists had a very poor idea
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anatómovia mali až donedávna veľmi chabé znalosti
06:10
of the structure of the lung until very recently.
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o štruktúre pľúc.
06:13
And I think that my mathematics,
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A myslím, že moja matematika,
06:15
surprisingly enough,
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prekvapujúco,
06:17
has been of great help
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veľmi pomohla
06:19
to the surgeons
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chirurgom,
06:21
studying lung illnesses
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ktorí študujú choroby pľúc
06:23
and also kidney illnesses,
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a tiež ochorenia obličiek,
06:25
all these branching systems,
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všetkých týchto vetviacich sa systémov,
06:27
for which there was no geometry.
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pre ktoré nebola žiadna geometria.
06:30
So I found myself, in other words,
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Inými slovami, ocitol som sa
06:32
constructing a geometry,
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v konštruovaní geometrie,
06:34
a geometry of things which had no geometry.
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geometrie vecí, ktoré nemali geometriu.
06:37
And a surprising aspect of it
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Prekvapujúcim aspektom toho
06:39
is that very often, the rules of this geometry
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je, že veľmi často, pravidlá tejto geometrie
06:42
are extremely short.
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sú extrémne krátke.
06:44
You have formulas that long.
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Máte takéto krátke vzorce.
06:46
And you crank it several times.
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Párkrát ich pretočíte.
06:48
Sometimes repeatedly: again, again, again,
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Niekedy opakovane znova, znova, znova.
06:50
the same repetition.
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To isté opakovanie.
06:52
And at the end, you get things like that.
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A na koniec dostanete veci ako toto.
06:54
This cloud is completely,
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Tento oblak je úplne,
06:56
100 percent artificial.
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100% umelý.
06:59
Well, 99.9.
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Nuž, 99,9.
07:01
And the only part which is natural
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Jediná prirodzená časť
07:03
is a number, the roughness of the cloud,
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je číslo, drsnosť oblaku,
07:05
which is taken from nature.
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ktoré je prevzaté z prírody.
07:07
Something so complicated like a cloud,
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Niečo tak komplikované ako oblak,
07:09
so unstable, so varying,
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tak nestále, tak premenlivé,
07:11
should have a simple rule behind it.
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by malo mať v pozadí jednoduché pravidlo.
07:14
Now this simple rule
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Toto jednoduché pravidlo
07:17
is not an explanation of clouds.
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nie je vysvetlením oblakov.
07:20
The seer of clouds had to
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Pozorovateľ oblakov to musel
07:22
take account of it.
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vziať do úvahy.
07:24
I don't know how much advanced
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Neviem nakoľko pokročilé
07:27
these pictures are. They're old.
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sú tieto obrázky, sú staré.
07:29
I was very much involved in it,
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Bol som súčasťou tejto oblasti,
07:31
but then turned my attention to other phenomena.
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ale potom som obrátil pozornosť na iné fenomény.
07:34
Now, here is another thing
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Je ďalšia vec,
07:36
which is rather interesting.
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ktorá je pomerne zaujímavá.
07:39
One of the shattering events
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Jedna z šokujúcich udalostí
07:41
in the history of mathematics,
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v histórii matematiky,
07:43
which is not appreciated by many people,
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nedoceňovaná mnohými ľuďmi,
07:46
occurred about 130 years ago,
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sa vyskytla asi pred 130 rokmi,
07:48
145 years ago.
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pred 145 rokmi.
07:50
Mathematicians began to create
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Matematici začali tvoriť
07:52
shapes that didn't exist.
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tvary, ktoré neexistovali.
07:54
Mathematicians got into self-praise
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Matematici začali chváliť sami seba
07:57
to an extent which was absolutely amazing,
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až do úplne neuveriteľných rozmerov,
07:59
that man can invent things
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za to, že človek dokáže vymyslieť veci,
08:01
that nature did not know.
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ktoré príroda nepoznala.
08:03
In particular, it could invent
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Napríklad, vytvorili
08:05
things like a curve which fills the plane.
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veci ako krivku, ktorá vyplní rovinu.
08:08
A curve's a curve, a plane's a plane,
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Krivka je krivka, rovina je rovina
08:10
and the two won't mix.
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a to nejde dokopy.
08:12
Well, they do mix.
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Nuž, ide.
08:14
A man named Peano
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Muž, Peano
08:16
did define such curves,
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definoval také krivky
08:18
and it became an object of extraordinary interest.
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a stalo sa to objektom nevídaného záujmu.
08:21
It was very important, but mostly interesting
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Bolo to veľmi dôležité, ale hlavne zaujímavé,
08:24
because a kind of break,
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lebo to bol istý druh zlomu,
08:26
a separation between
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separácia medzi
08:28
the mathematics coming from reality, on the one hand,
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matematikou vychádzajúcej z reality na jednej strane
08:31
and new mathematics coming from pure man's mind.
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a novou matematikou, vychádzajúcej z čistej ľudskej mysle.
08:34
Well, I was very sorry to point out
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Nuž, veľmi nerád som vtedy pripomenul,
08:37
that the pure man's mind
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že čistá ľudská myseľ
08:39
has, in fact,
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vlastne,
08:41
seen at long last
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konečne uvidela,
08:43
what had been seen for a long time.
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čo bolo k videniu už dlhý čas.
08:45
And so here I introduce something,
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Takže teraz niečo predstavím,
08:47
the set of rivers of a plane-filling curve.
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súbor riek krivky vypĺňajúcej rovinu.
08:50
And well,
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A vlastne,
08:52
it's a story unto itself.
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je to príbeh sám o sebe.
08:54
So it was in 1875 to 1925,
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Takže, bolo to v rokoch 1875 až 1925,
08:57
an extraordinary period
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výnimočné obdobie,
08:59
in which mathematics prepared itself to break out from the world.
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v ktorej sa matematici pripravovali uniknúť z tohto sveta.
09:02
And the objects which were used
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A objekty, ktoré používali
09:04
as examples, when I was
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ako príklady -- keď som bol
09:06
a child and a student, as examples
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dieťa a študent --
09:08
of the break between mathematics
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rozchodu medzi matematikou
09:11
and visible reality --
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a viditeľnou realitou --
09:13
those objects,
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tieto objekty,
09:15
I turned them completely around.
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som úplne otočil.
09:17
I used them for describing
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Použil som ich na popísanie
09:19
some of the aspects of the complexity of nature.
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niektorých aspektov komplexity prírody.
09:22
Well, a man named Hausdorff in 1919
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Nuž, istý Hausdorff v roku 1919
09:25
introduced a number which was just a mathematical joke,
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predstavil číslo, ktoré bolo len matematickým vtipom.
09:28
and I found that this number
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A prišiel som na to, že toto číslo
09:30
was a good measurement of roughness.
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je dobrým meradlom drsnosti.
09:32
When I first told it to my friends in mathematics
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Keď som to prvýkrát povedal priateľom matematikom,
09:34
they said, "Don't be silly. It's just something [silly]."
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vraveli, "Nebuď blázon. To je len nejaká blbosť."
09:37
Well actually, I was not silly.
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Nuž, nakoniec, nebola to blbosť.
09:40
The great painter Hokusai knew it very well.
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Veľký maliar Hokusai to dobre poznal.
09:43
The things on the ground are algae.
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Tie veci na zemi sú riasy.
09:45
He did not know the mathematics; it didn't yet exist.
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Nepoznal tú matematiku, ešte neexistovala.
09:48
And he was Japanese who had no contact with the West.
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A bol Japonec, teda nemal kontakt so západom.
09:51
But painting for a long time had a fractal side.
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Ale maliarstvo malo po dlhý čas fraktálovú stránku.
09:54
I could speak of that for a long time.
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Mohol by som o tom dlho hovoriť.
09:56
The Eiffel Tower has a fractal aspect.
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Eiffelova veža má fraktálový aspekt.
09:59
I read the book that Mr. Eiffel wrote about his tower,
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A čítal som knihu, ktorú o tej veži napísal pán Eiffel.
10:02
and indeed it was astonishing how much he understood.
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A bolo to naozaj úžasné, ako tomu rozumel.
10:05
This is a mess, mess, mess, Brownian loop.
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Toto je chaos, chaos, chaos, Brownova slučka.
10:08
One day I decided --
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Jedného dňa som sa rozhodol, že
10:10
halfway through my career,
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na polceste mojej kariéry
10:12
I was held by so many things in my work --
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som bol zväzovaný toľkými vecami v mojej práci,
10:15
I decided to test myself.
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rozhodol som sa otestovať sám seba.
10:18
Could I just look at something
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Dokážem sa len tak pozrieť na niečo,
10:20
which everybody had been looking at for a long time
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na čo všetci pozerali po dlhý čas
10:23
and find something dramatically new?
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a nájsť niečo úplne nové?
10:26
Well, so I looked at these
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Nuž, pozrel som sa na tieto
10:29
things called Brownian motion -- just goes around.
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veci, zvané Brownov pohyb -- iba to chodí dokola.
10:32
I played with it for a while,
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Nejaký čas som sa s tým hral
10:34
and I made it return to the origin.
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a prinútil som to vrátiť sa na začiatok.
10:37
Then I was telling my assistant,
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Potom som povedal asistentovi,
10:39
"I don't see anything. Can you paint it?"
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"Nič nevidím. Môžete to nakresliť?"
10:41
So he painted it, which means
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Nakreslil to, čo znamená,
10:43
he put inside everything. He said:
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že to vyplnil. Povedal:
10:45
"Well, this thing came out ..." And I said, "Stop! Stop! Stop!
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"Nuž, toto z toho vyšlo.." A vravím, "Stačí! Stačí! Stačí!
10:48
I see; it's an island."
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Vidím, je to ostrov."
10:51
And amazing.
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A pekný.
10:53
So Brownian motion, which happens to have
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Takže Brownov pohyb, ktorý má
10:55
a roughness number of two, goes around.
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číslo hrubosti dva, ide dookola.
10:58
I measured it, 1.33.
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Odmeral som to, 1,33.
11:00
Again, again, again.
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Znova, znova a znova.
11:02
Long measurements, big Brownian motions,
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Dlhé merania, veľké Brownove pohyby,
11:04
1.33.
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1,33.
11:06
Mathematical problem: how to prove it?
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Matematický problém: ako to dokázať?
11:09
It took my friends 20 years.
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Mojim priateľom to zabralo 20 rokov.
11:12
Three of them were having incomplete proofs.
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Traja z nich mali čiastočné dôkazy.
11:15
They got together, and together they had the proof.
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Dali sa dokopy a spolu mali dôkaz.
11:19
So they got the big [Fields] medal in mathematics,
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Dostali veľkú (Fieldsovu) medailu za matematiku,
11:22
one of the three medals that people have received
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jednu z troch medailí, ktoré dostali ľudia
11:24
for proving things which I've seen
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dokazujúci veci, ktoré som videl,
11:27
without being able to prove them.
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ale nebol schopný dokázať.
11:30
Now everybody asks me at one point or another,
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Každý sa ma z času na čas pýta,
11:33
"How did it all start?
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"Ako to začalo?
11:35
What got you in that strange business?"
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Čo vás priviedlo k týmto divným témam?"
11:38
What got you to be,
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Čo zo mňa spravilo
11:40
at the same time, a mechanical engineer,
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zároveň mechanickýho inžiniera,
11:42
a geographer
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geografa
11:44
and a mathematician and so on, a physicist?
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a matematika a tak ďalej, fyzika?
11:46
Well actually I started, oddly enough,
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Nuž, vlastne som začal, dosť čudné,
11:49
studying stock market prices.
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študovať ceny na finančnom trhu.
11:51
And so here
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A tak,
11:53
I had this theory,
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mal som teóriu,
11:56
and I wrote books about it --
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napísal som o tom knihy,
11:58
financial prices increments.
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Prírastky finančných cien.
12:00
To the left you see data over a long period.
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Naľavo vidíte dáta z dlhšieho obdobia.
12:02
To the right, on top,
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Napravo, hore,
12:04
you see a theory which is very, very fashionable.
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vidíte veľmi, veľmi elegantnú teóriu.
12:07
It was very easy, and you can write many books very fast about it.
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Bolo to veľmi ľahké a môžete o tom veľmi rýchlo napísať veľa kníh.
12:10
(Laughter)
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(Smiech)
12:12
There are thousands of books on that.
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Sú o tom tisícky kníh.
12:15
Now compare that with real price increments.
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Porovnajte to so skutočnými zmenami ceny.
12:18
Where are real price increments?
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A kde sú skutočné zmeny cien?
12:20
Well, these other lines
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Nuž, tieto ostatné čiary
12:22
include some real price increments
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obsahujú niektoré reálne zmeny cien
12:24
and some forgery which I did.
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a pár mojich podvrhov.
12:26
So the idea there was
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Takže idea bola,
12:28
that one must be able to -- how do you say? --
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že sa musí dať -- ako sa to povie --
12:30
model price variation.
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modelovať cenové variácie.
12:33
And it went really well 50 years ago.
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A celkom to fungovalo pred 50 rokmi.
12:36
For 50 years, people were sort of pooh-poohing me
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50 rokov si ma ľudia doberali,
12:39
because they could do it much, much easier.
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lebo oni to vedeli oveľa oveľa ľahšie.
12:41
But I tell you, at this point, people listened to me.
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Ale potom ma začali počúvať.
12:44
(Laughter)
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(Smiech)
12:46
These two curves are averages:
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Tieto dve krivky sú priemery.
12:48
Standard & Poor, the blue one;
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Standard & Poor, modrá.
12:50
and the red one is Standard & Poor's
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A červená je Standard & Poor,
12:52
from which the five biggest discontinuities
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bez piatich najväčších
12:55
are taken out.
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nepravidelností.
12:57
Now discontinuities are a nuisance,
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Nepravidelnosti sú otrava.
12:59
so in many studies of prices,
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A preto sú v mnohých štúdiách cien,
13:02
one puts them aside.
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vylučované.
13:04
"Well, acts of God.
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"Nuž, Božie zásahy.
13:06
And you have the little nonsense which is left.
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A zostane vám tá trocha nezmyslov.
13:09
Acts of God." In this picture,
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Božie zásahy." Na tomto obrázku
13:12
five acts of God are as important as everything else.
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päť Božích zásahov je tak dôležitých ako všetko ostatné.
13:15
In other words,
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Inými slovami,
13:17
it is not acts of God that we should put aside.
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nie sú to tie Božie zásahy, ktoré by sme mali odložiť.
13:19
That is the meat, the problem.
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To je podstata, ten problém.
13:22
If you master these, you master price,
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Ak zvládnete tie, zvládnete ceny.
13:25
and if you don't master these, you can master
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A ak ich nezvládnete, môžete zvládnuť
13:27
the little noise as well as you can,
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ten šum ako len chcete,
13:29
but it's not important.
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ale bude to jedno.
13:31
Well, here are the curves for it.
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Nuž, tu sú k tomu krivky.
13:33
Now, I get to the final thing, which is the set
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Teraz sa dostávam k hlavnej veci, čo je množina,
13:35
of which my name is attached.
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ktorá nesie moje meno.
13:37
In a way, it's the story of my life.
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Svojim spôsobom je to príbeh môjho života.
13:39
My adolescence was spent
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Svoju mladosť som strávil
13:41
during the German occupation of France.
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počas nemeckej okupácie Francúzska.
13:43
Since I thought that I might
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A keďže som si myslel, že
13:46
vanish within a day or a week,
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možno neprežijem deň alebo týždeň,
13:49
I had very big dreams.
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mal som veľké sny.
13:52
And after the war,
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A po vojne,
13:54
I saw an uncle again.
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som znova uvidel strýka.
13:56
My uncle was a very prominent mathematician, and he told me,
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Strýko bol slávny matematik a povedal mi,
13:58
"Look, there's a problem
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"Pozri, je taký problém,
14:00
which I could not solve 25 years ago,
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ktorý som nedokázal vyriešiť pred 25 rokmi,
14:02
and which nobody can solve.
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a ktorý nikto nevie vyriešiť.
14:04
This is a construction of a man named [Gaston] Julia
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Toto je konštrukcia pánov (Gaston) Julia
14:06
and [Pierre] Fatou.
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a (Pierre) Fatou.
14:08
If you could
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Ak by si našiel
14:10
find something new, anything,
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niečo nové, hocičo,
14:12
you will get your career made."
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postavíš si na tom kariéru."
14:14
Very simple.
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Veľmi jednoduché.
14:16
So I looked,
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Tak som sa na to pozrel
14:18
and like the thousands of people that had tried before,
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a ako tisíce ľudí, ktorí to skúšali predtým,
14:20
I found nothing.
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nič som nenašiel.
14:23
But then the computer came,
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A potom prišiel počítač.
14:25
and I decided to apply the computer,
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Rozhodol som sa použiť počítač,
14:27
not to new problems in mathematics --
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nie na nové problémy v matematike --
14:30
like this wiggle wiggle, that's a new problem --
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ako tieto čáry-máry, to je nový problém --
14:32
but to old problems.
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ale na staré problémy.
14:34
And I went from what's called
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A prešiel som od toho, čo voláme
14:36
real numbers, which are points on a line,
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reálne čísla, ktoré sú body na priamke,
14:38
to imaginary, complex numbers,
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na imaginárne, komplexné čísla,
14:40
which are points on a plane,
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ktoré sú bodmi na rovine,
14:42
which is what one should do there,
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a to sa malo skúsiť.
14:44
and this shape came out.
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A vyšiel z toho tento tvar.
14:46
This shape is of an extraordinary complication.
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Tento tvar má výnimočnú zložitosť.
14:49
The equation is hidden there,
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Rovnica je skrytá tu,
14:51
z goes into z squared, plus c.
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z sa rovná z na druhú plus c.
14:54
It's so simple, so dry.
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Je to také jednoduché, suché.
14:56
It's so uninteresting.
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Je to tak nezaujímavé.
14:58
Now you turn the crank once, twice:
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Otočíte kľukou raz, dvakrát,
15:01
twice,
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dvakrát
15:04
marvels come out.
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a začnú sa divy.
15:06
I mean this comes out.
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Vypadne z toho toto.
15:08
I don't want to explain these things.
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Nechcem to vysvetľovať.
15:10
This comes out. This comes out.
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Vyjde toto. Vyjde toto.
15:12
Shapes which are of such complication,
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Tvary, ktoré sú tak zložité,
15:14
such harmony and such beauty.
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tak harmonické a také krásne.
15:17
This comes out
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Toto z toho vychádza
15:19
repeatedly, again, again, again.
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opakovane, znova, znova, znova.
15:21
And that was one of my major discoveries,
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A to bol jeden z mojich veľkých objavov,
15:23
to find that these islands were the same
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objaviť, že tieto ostrovy boli také isté,
15:25
as the whole big thing, more or less.
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ako celá tá veľká vec, viac-menej.
15:27
And then you get these
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A potom dostanete
15:29
extraordinary baroque decorations all over the place.
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všade tieto výnimočné barokové ozdoby.
15:32
All that from this little formula,
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To všetko z tohto malého vzorca,
15:35
which has whatever, five symbols in it.
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ktorý má, povedzme, päť symbolov.
15:38
And then this one.
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A potom tento.
15:40
The color was added for two reasons.
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Farba bola pridaná z dvoch dôvodov.
15:42
First of all, because these shapes
386
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Po prvé, lebo tieto tvary
15:44
are so complicated
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sú také zložité,
15:47
that one couldn't make any sense of the numbers.
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že by z čísel nešlo nič pochopiť.
15:50
And if you plot them, you must choose some system.
389
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A ak ich nakreslíte, musíte si vybrať nejaký systém.
15:53
And so my principle has been
390
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A tak mojim princípom bolo
15:55
to always present the shapes
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vždy prezentovať tvary
15:58
with different colorings
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rôznym zafarbením,
16:00
because some colorings emphasize that,
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lebo niektoré zafarbenia zdôrazňujú niečo
16:02
and others it is that or that.
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a iné zas to alebo ono.
16:04
It's so complicated.
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Je to také zložité.
16:06
(Laughter)
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(Smiech)
16:08
In 1990, I was in Cambridge, U.K.
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V roku 1990 som bol v Cambridge, vo Veľkej Británii,
16:10
to receive a prize from the university,
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prebrať cenu od univerzity.
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and three days later,
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A o tri dni neskôr
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a pilot was flying over the landscape and found this thing.
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pilot prelietal krajinou a našiel túto vec.
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So where did this come from?
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Odkiaľ sa to tam vzalo?
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Obviously, from extraterrestrials.
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Samozrejme, mimozemšťania.
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(Laughter)
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(Smiech)
16:25
Well, so the newspaper in Cambridge
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Nuž, noviny v Cambridge
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published an article about that "discovery"
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uverejnili článok o tomto "náleze"
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and received the next day
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a na ďalší deň dostali
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5,000 letters from people saying,
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5000 listov od ľudí, čo vraveli,
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"But that's simply a Mandelbrot set very big."
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"Ale to je proste veľká Mandelbrotova množina."
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Well, let me finish.
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Nuž, na záver.
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This shape here just came
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Tento tvar vzišiel
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out of an exercise in pure mathematics.
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z čistej matematiky.
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Bottomless wonders spring from simple rules,
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Bezodné divy prýštia z jednoduchých pravidiel,
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which are repeated without end.
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ktoré sa opakujú bez konca.
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Thank you very much.
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Ďakujem veľmi pekne.
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(Applause)
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(Potlesk)
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