TED2010
Benoit Mandelbrot: Fractals and the art of roughness
Benoit Mandelbrot: Les fractals i l'art de la rugositat
Filmed:
Readability: 3.2
1,448,555 views
Al TED2010, una llegenda de les matemàtiques, en Benoit Mandelbrot, exposa el tema que ja va presentar al TED el 1984 -- la complexitat extrema de la rugositat, i com la matemàtica fractal pot trobar ordre en estructures que semblen increïblement complicades.
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.
0
0
2000
Moltes gràcies.
00:17
Please excuse me for sitting; I'm very old.
1
2000
3000
Perdoneu-me per seure; sóc molt gran.
00:20
(Laughter)
2
5000
2000
(Rialles)
00:22
Well, the topic I'm going to discuss
3
7000
2000
Bé, el tema que vaig a exposar
00:24
is one which is, in a certain sense, very peculiar
4
9000
3000
és un que, en certa manera, és ben peculiar
00:27
because it's very old.
5
12000
2000
perquè és molt antic.
00:29
Roughness is part of human life
6
14000
3000
La rugositat és part de la vida humana
00:32
forever and forever,
7
17000
2000
des de sempre.
00:34
and ancient authors have written about it.
8
19000
3000
Antics autors ja en van parlar.
00:37
It was very much uncontrollable,
9
22000
2000
No es podia dominar.
00:39
and in a certain sense,
10
24000
2000
I en certa manera,
00:41
it seemed to be the extreme of complexity,
11
26000
3000
semblava ser al capdamunt de la complexitat.
00:44
just a mess, a mess and a mess.
12
29000
2000
un desori, un desori i un desori.
00:46
There are many different kinds of mess.
13
31000
2000
Hi ha moltes menes diferents de desoris.
00:48
Now, in fact,
14
33000
2000
Ara, de fet,
00:50
by a complete fluke,
15
35000
2000
per pura xamba,
00:52
I got involved many years ago
16
37000
3000
fa molts anys em vaig veure embolicat
00:55
in a study of this form of complexity,
17
40000
3000
en un estudi d'aquest tipus de complexitat.
00:58
and to my utter amazement,
18
43000
2000
I, contra tot pronòstic,
01:00
I found traces --
19
45000
2000
vaig trobar indicis --
01:02
very strong traces, I must say --
20
47000
2000
indicis prou clars, haig de dir --
01:04
of order in that roughness.
21
49000
3000
d'ordre en aquella rugositat.
01:07
And so today, I would like to present to you
22
52000
2000
I així avui, voldria presentar-vos
01:09
a few examples
23
54000
2000
uns quants exemples
01:11
of what this represents.
24
56000
2000
del que això vol dir.
01:13
I prefer the word roughness
25
58000
2000
Prefereixo el mot rugositat
01:15
to the word irregularity
26
60000
2000
al mot irregularitat
01:17
because irregularity --
27
62000
2000
perquè irregularitat --
01:19
to someone who had Latin
28
64000
2000
per a algú com jo que vaig estudiar llatí
01:21
in my long-past youth --
29
66000
2000
en la meva joventut, ja fa molt temps --
01:23
means the contrary of regularity.
30
68000
2000
significa el contrari de la regularitat.
01:25
But it is not so.
31
70000
2000
Però no és així.
01:27
Regularity is the contrary of roughness
32
72000
3000
La regularitat és el contrari de la rugositat
01:30
because the basic aspect of the world
33
75000
2000
perquè l'aspecte bàsic del món
01:32
is very rough.
34
77000
2000
és molt rugós.
01:34
So let me show you a few objects.
35
79000
3000
Deixeu-me mostrar-vos alguns objectes.
01:37
Some of them are artificial.
36
82000
2000
Alguns són artificials.
01:39
Others of them are very real, in a certain sense.
37
84000
3000
Altres són ben reals, d'alguna manera.
01:42
Now this is the real. It's a cauliflower.
38
87000
3000
Això és real. És una coliflor.
01:45
Now why do I show a cauliflower,
39
90000
3000
Per què us ensenyo una coliflor,
01:48
a very ordinary and ancient vegetable?
40
93000
3000
una verdura ben normal i antiga?
01:51
Because old and ancient as it may be,
41
96000
3000
Perquè tan antiga cóm és,
01:54
it's very complicated and it's very simple,
42
99000
3000
és molt complicada i és molt senzilla
01:57
both at the same time.
43
102000
2000
les dues coses alhora.
01:59
If you try to weigh it -- of course it's very easy to weigh it,
44
104000
3000
Si la voleu pesar, és ben senzilla de pesar.
02:02
and when you eat it, the weight matters --
45
107000
3000
I quan la mengeu, el pes és important.
02:05
but suppose you try to
46
110000
3000
Però suposeu que voleu
02:08
measure its surface.
47
113000
2000
mesurar-ne la superfície.
02:10
Well, it's very interesting.
48
115000
2000
Bé, és molt interessant.
02:12
If you cut, with a sharp knife,
49
117000
3000
Si talleu, amb un ganivet esmolat,
02:15
one of the florets of a cauliflower
50
120000
2000
un dels brots de la coliflor
02:17
and look at it separately,
51
122000
2000
i el mireu separadament,
02:19
you think of a whole cauliflower, but smaller.
52
124000
3000
us recorda la coliflor sencera, però més petita.
02:22
And then you cut again,
53
127000
2000
I la talleu un altre cop,
02:24
again, again, again, again, again, again, again, again,
54
129000
3000
i un altre, i un altre, i un altre, i un altre, i un altre.
02:27
and you still get small cauliflowers.
55
132000
2000
I encara tindreu petites coliflors.
02:29
So the experience of humanity
56
134000
2000
Així, l'experiència de la humanitat
02:31
has always been that there are some shapes
57
136000
3000
sempre ha estat que hi ha algunes formes
02:34
which have this peculiar property,
58
139000
2000
que tenen aquesta propietat singular,
02:36
that each part is like the whole,
59
141000
3000
que cada part és com el tot,
02:39
but smaller.
60
144000
2000
però més petita.
02:41
Now, what did humanity do with that?
61
146000
3000
I què en va fer la humanitat d'això?
02:44
Very, very little.
62
149000
3000
Ben poca cosa.
02:47
(Laughter)
63
152000
3000
(Rialles)
02:50
So what I did actually is to
64
155000
3000
Doncs el que vaig fer va ser
02:53
study this problem,
65
158000
3000
estudiar aquest problema,
02:56
and I found something quite surprising.
66
161000
3000
i em vaig trobar amb una cosa prou sorprenent.
02:59
That one can measure roughness
67
164000
3000
Que un pot mesurar la rugositat
03:02
by a number, a number,
68
167000
3000
amb un nombre, un nombre,
03:05
2.3, 1.2 and sometimes much more.
69
170000
3000
"2,3", "1,2" i a vegades molt més.
03:08
One day, a friend of mine,
70
173000
2000
Un dia, un amic meu,
03:10
to bug me,
71
175000
2000
per provocar-me,
03:12
brought a picture and said,
72
177000
2000
va portar una foto, i em va dir,
03:14
"What is the roughness of this curve?"
73
179000
2000
"Quina és la rugositat d'aquesta corba?"
03:16
I said, "Well, just short of 1.5."
74
181000
3000
Vaig dir, "Bé, no arriba a 1,5"
03:19
It was 1.48.
75
184000
2000
Era 1,48.
03:21
Now, it didn't take me any time.
76
186000
2000
No m'hi va costar gaire.
03:23
I've been looking at these things for so long.
77
188000
2000
He estat mirant aquestes coses durant molt de temps.
03:25
So these numbers are the numbers
78
190000
2000
I aquests nombres són els nombres
03:27
which denote the roughness of these surfaces.
79
192000
3000
que indiquen la rugositat d'aquestes superfícies.
03:30
I hasten to say that these surfaces
80
195000
2000
M'afanyo a dir que aquestes superfícies
03:32
are completely artificial.
81
197000
2000
són artificials del tot.
03:34
They were done on a computer,
82
199000
2000
Estàn fetes amb un ordinador.
03:36
and the only input is a number,
83
201000
2000
I que l'única dada és un nombre.
03:38
and that number is roughness.
84
203000
3000
I que el nombre és la rugositat.
03:41
So on the left,
85
206000
2000
Així a l'esquerra,
03:43
I took the roughness copied from many landscapes.
86
208000
3000
agafo la rugositat semblant a la de molts paisatges.
03:46
To the right, I took a higher roughness.
87
211000
3000
A l'esquerra, agafo una rugositat més gran.
03:49
So the eye, after a while,
88
214000
2000
I l'ull, al cap de poc,
03:51
can distinguish these two very well.
89
216000
3000
pot distingir-ne les dues molt bé.
03:54
Humanity had to learn about measuring roughness.
90
219000
2000
La humanitat havia d'aprendre a mesurar la rugositat.
03:56
This is very rough, and this is sort of smooth, and this perfectly smooth.
91
221000
3000
Això és molt rugós, i aixó és una mica suau, i aixó és totalment llis.
03:59
Very few things are very smooth.
92
224000
3000
Molt poques coses són ben suaus.
04:03
So then if you try to ask questions:
93
228000
3000
Quan intentes preguntar-te:
04:06
"What's the surface of a cauliflower?"
94
231000
2000
quina és la superfície d'una coliflor?
04:08
Well, you measure and measure and measure.
95
233000
3000
Bé, mesures i mesures i mesures.
04:11
Each time you're closer, it gets bigger,
96
236000
3000
I quan més a la vora arribes es fa més gran,
04:14
down to very, very small distances.
97
239000
2000
fins arribar a distàncies molt i molt petites.
04:16
What's the length of the coastline
98
241000
2000
Quina és la llargada de la riba
04:18
of these lakes?
99
243000
2000
d'aquests llacs?
04:20
The closer you measure, the longer it is.
100
245000
3000
Com més a la vora ho mesures, més llarg resulta ser.
04:23
The concept of length of coastline,
101
248000
2000
El concepte de la llargada de la riba,
04:25
which seems to be so natural
102
250000
2000
que sembla tan natural
04:27
because it's given in many cases,
103
252000
2000
perquè se'ns dóna moltes vegades,
04:29
is, in fact, complete fallacy; there's no such thing.
104
254000
3000
és, de fet, una completa fal·làcia; no existeix.
04:32
You must do it differently.
105
257000
3000
S'ha de fer d'una altra manera.
04:35
What good is that, to know these things?
106
260000
2000
Per què és bo saber aquestes coses?
04:37
Well, surprisingly enough,
107
262000
2000
Bé, sorprèn força perquè
04:39
it's good in many ways.
108
264000
2000
és bo per molts motius.
04:41
To begin with, artificial landscapes,
109
266000
2000
Per començar, els paisatges artificials,
04:43
which I invented sort of,
110
268000
2000
que d'alguna manera vaig inventar,
04:45
are used in cinema all the time.
111
270000
3000
s'utilitzen al cinema molt sovint.
04:48
We see mountains in the distance.
112
273000
2000
Veiem muntanyes en la distància.
04:50
They may be mountains, but they may be just formulae, just cranked on.
113
275000
3000
Poden ser muntanyes, però poden ser generades només per fórmules.
04:53
Now it's very easy to do.
114
278000
2000
Ara és molt fàcil de fer.
04:55
It used to be very time-consuming, but now it's nothing.
115
280000
3000
Abans calia molt de temps, però ara no costa res.
04:58
Now look at that. That's a real lung.
116
283000
3000
Ara mireu això. És un pulmó de veritat.
05:01
Now a lung is something very strange.
117
286000
2000
Bé, un pulmó és una cosa ben rara.
05:03
If you take this thing,
118
288000
2000
Si agafes això,
05:05
you know very well it weighs very little.
119
290000
3000
ja sabeu molt bé que pesa ben poc.
05:08
The volume of a lung is very small,
120
293000
2000
El volum d'un pulmó és molt petit.
05:10
but what about the area of the lung?
121
295000
3000
Però quina és l'àrea del pulmó?
05:13
Anatomists were arguing very much about that.
122
298000
3000
Els anatomistes segueixen discutint molt sobre això.
05:16
Some say that a normal male's lung
123
301000
3000
Alguns diuen que un pulmó normal d'un home
05:19
has an area of the inside
124
304000
2000
té una àrea com l'interior
05:21
of a basketball [court].
125
306000
2000
d'una pista de bàsquet.
05:23
And the others say, no, five basketball [courts].
126
308000
3000
Altres diuen, no, de cinc pistes de bàsquet.
05:27
Enormous disagreements.
127
312000
2000
Grans diferències.
05:29
Why so? Because, in fact, the area of the lung
128
314000
3000
Perquè passa això? Perquè, de fet, l'àrea del pulmó
05:32
is something very ill-defined.
129
317000
2000
és una cosa gens ben definida.
05:35
The bronchi branch, branch, branch
130
320000
3000
Els bronquis es ramifiquen, es ramifiquen, es ramifiquen.
05:38
and they stop branching,
131
323000
3000
I paren de ramificar-se,
05:41
not because of any matter of principle,
132
326000
3000
no per cap qüestió de principis,
05:44
but because of physical considerations:
133
329000
3000
sinó per consideracions físiques,
05:47
the mucus, which is in the lung.
134
332000
3000
per la mucositat que hi ha al pulmó.
05:50
So what happens is that in a way
135
335000
2000
El que passa és que aquesta és la manera
05:52
you have a much bigger lung,
136
337000
2000
de tenir un pulmó molt més gran,
05:54
but it branches and branches
137
339000
2000
i es ramifica i ramifica,
05:56
down to distances about the same for a whale, for a man
138
341000
3000
fins a mides molt semblants per a una balena, un home
05:59
and for a little rodent.
139
344000
2000
o un petit rosegador.
06:02
Now, what good is it to have that?
140
347000
3000
Ara bé, per a què és bo que això sigui així?
06:05
Well, surprisingly enough, amazingly enough,
141
350000
2000
Bé, és ben sorprenent i ben curiós,
06:07
the anatomists had a very poor idea
142
352000
3000
que els anatomistes no en tinguessin ni idea
06:10
of the structure of the lung until very recently.
143
355000
3000
de l'estructura del pulmó fins fa ben poc.
06:13
And I think that my mathematics,
144
358000
2000
I penso que les meves matemàtiques,
06:15
surprisingly enough,
145
360000
2000
de forma sorprenent,
06:17
has been of great help
146
362000
2000
han ajudat molt
06:19
to the surgeons
147
364000
2000
els cirurgians
06:21
studying lung illnesses
148
366000
2000
que estudien les malaties pulmonars
06:23
and also kidney illnesses,
149
368000
2000
i també les dels ronyons.
06:25
all these branching systems,
150
370000
2000
Tots aquests sistemes ramificats,
06:27
for which there was no geometry.
151
372000
3000
per als quals no hi havia cap geometria.
06:30
So I found myself, in other words,
152
375000
2000
I em vaig trobar, en altres paraules,
06:32
constructing a geometry,
153
377000
2000
construïnt una geometria,
06:34
a geometry of things which had no geometry.
154
379000
3000
una geometria de les coses que no tenien geometria.
06:37
And a surprising aspect of it
155
382000
2000
I un aspecte curiós d'això
06:39
is that very often, the rules of this geometry
156
384000
3000
és que molt sovint, les regles d'aquesta geometria
06:42
are extremely short.
157
387000
2000
són molt poques.
06:44
You have formulas that long.
158
389000
2000
Tens fórmules d'aquesta mida.
06:46
And you crank it several times.
159
391000
2000
I les utilitzem moltes vegades.
06:48
Sometimes repeatedly: again, again, again,
160
393000
2000
De vegades repetidament, un cop i un altre, i un altre.
06:50
the same repetition.
161
395000
2000
La mateixa repetició.
06:52
And at the end, you get things like that.
162
397000
2000
I al final obtens coses com aquesta.
06:54
This cloud is completely,
163
399000
2000
Aquest núvol és completament,
06:56
100 percent artificial.
164
401000
3000
100 % artificial.
06:59
Well, 99.9.
165
404000
2000
Bé, 99,9 %.
07:01
And the only part which is natural
166
406000
2000
I l'unica cosa que és natural
07:03
is a number, the roughness of the cloud,
167
408000
2000
és un nombre, la rugositat del núvol,
07:05
which is taken from nature.
168
410000
2000
que l'hem pres de la natura.
07:07
Something so complicated like a cloud,
169
412000
2000
Una cosa tan complicada com un núvol,
07:09
so unstable, so varying,
170
414000
2000
tan inestable, tan canviant,
07:11
should have a simple rule behind it.
171
416000
3000
havia de tenir una senzilla regla al darrere.
07:14
Now this simple rule
172
419000
3000
Ara bé, aquesta regla senzilla
07:17
is not an explanation of clouds.
173
422000
3000
no explica els núvols.
07:20
The seer of clouds had to
174
425000
2000
L'observador de núvols hauria de
07:22
take account of it.
175
427000
2000
tenir-ho en compte.
07:24
I don't know how much advanced
176
429000
3000
No sé com són d'avançades
07:27
these pictures are. They're old.
177
432000
2000
aquestes fotos, ja són antigues.
07:29
I was very much involved in it,
178
434000
2000
Vaig estar molt involucrat en això,
07:31
but then turned my attention to other phenomena.
179
436000
3000
però llavors em vaig fixar en un altre fenomen.
07:34
Now, here is another thing
180
439000
2000
Bé, hi ha una altra cosa
07:36
which is rather interesting.
181
441000
3000
que és ben interessant.
07:39
One of the shattering events
182
444000
2000
Un d'aquests moments clau
07:41
in the history of mathematics,
183
446000
2000
en la història de les matemàtiques,
07:43
which is not appreciated by many people,
184
448000
3000
que no és valorada per molta gent,
07:46
occurred about 130 years ago,
185
451000
2000
va passar fa uns 130 anys,
07:48
145 years ago.
186
453000
2000
145.
07:50
Mathematicians began to create
187
455000
2000
Els matemàtics van començar a crear
07:52
shapes that didn't exist.
188
457000
2000
formes que no existien.
07:54
Mathematicians got into self-praise
189
459000
3000
Els matemàtics es van felicitar,
07:57
to an extent which was absolutely amazing,
190
462000
2000
fins a un punt que era realment sorprenent,
07:59
that man can invent things
191
464000
2000
pel fet que l'home pogués inventar coses
08:01
that nature did not know.
192
466000
2000
que no hi havia a la natura.
08:03
In particular, it could invent
193
468000
2000
En particular, que poguessin inventar
08:05
things like a curve which fills the plane.
194
470000
3000
coses com una corba que omple el pla.
08:08
A curve's a curve, a plane's a plane,
195
473000
2000
Una corba és una corba i un pla és un pla,
08:10
and the two won't mix.
196
475000
2000
i no es poden barrejar.
08:12
Well, they do mix.
197
477000
2000
Doncs bé, s'hi barregen.
08:14
A man named Peano
198
479000
2000
Un home que s'anomenava Peano
08:16
did define such curves,
199
481000
2000
va definir aquesta mena de corbes,
08:18
and it became an object of extraordinary interest.
200
483000
3000
i va esdevenir l'objecte d'un interès extraordinari.
08:21
It was very important, but mostly interesting
201
486000
3000
Era molt important, però sobretot interessant
08:24
because a kind of break,
202
489000
2000
perquè era una mena de trencament,
08:26
a separation between
203
491000
2000
una separació entre
08:28
the mathematics coming from reality, on the one hand,
204
493000
3000
les matemàtiques provinents de la realitat d'una banda
08:31
and new mathematics coming from pure man's mind.
205
496000
3000
i les noves matemàtiques provinents purament de la ment humana.
08:34
Well, I was very sorry to point out
206
499000
3000
Bé, em va saber greu observar
08:37
that the pure man's mind
207
502000
2000
que la ment pura de l'home
08:39
has, in fact,
208
504000
2000
havia, de fet,
08:41
seen at long last
209
506000
2000
trigat molt a adonar-se
08:43
what had been seen for a long time.
210
508000
2000
d'allò que havia estat veient feia molt temps.
08:45
And so here I introduce something,
211
510000
2000
I ara us ensenyaré una cosa,
08:47
the set of rivers of a plane-filling curve.
212
512000
3000
el conjunt de rius d'una corba que omple un pla.
08:50
And well,
213
515000
2000
I bé,
08:52
it's a story unto itself.
214
517000
2000
és una història en si mateixa.
08:54
So it was in 1875 to 1925,
215
519000
3000
Això va ser entre 1875 i 1925,
08:57
an extraordinary period
216
522000
2000
un període extraordinari
08:59
in which mathematics prepared itself to break out from the world.
217
524000
3000
en què les matemàtiques es van preparar per separar-se del món.
09:02
And the objects which were used
218
527000
2000
I els objectes que s'empraven
09:04
as examples, when I was
219
529000
2000
com a exemples, quan era
09:06
a child and a student, as examples
220
531000
2000
nen i estudiant,
09:08
of the break between mathematics
221
533000
3000
del trencament entre les matemàtiques
09:11
and visible reality --
222
536000
2000
i la realitat visible --
09:13
those objects,
223
538000
2000
aquells objectes,
09:15
I turned them completely around.
224
540000
2000
els vaig capgirar completament.
09:17
I used them for describing
225
542000
2000
Jo els emprava per descriure
09:19
some of the aspects of the complexity of nature.
226
544000
3000
alguns aspectes de la complexitat de la natura.
09:22
Well, a man named Hausdorff in 1919
227
547000
3000
Bé, el 1919, un home anomenat Hausdorff
09:25
introduced a number which was just a mathematical joke,
228
550000
3000
va introduir un nombre que era un acudit matemàtic.
09:28
and I found that this number
229
553000
2000
I vaig trobar que aquest nombre
09:30
was a good measurement of roughness.
230
555000
2000
era una bona mesura de la rugositat.
09:32
When I first told it to my friends in mathematics
231
557000
2000
Quan ho vaig dir per primer cop als meus amics matemàtics
09:34
they said, "Don't be silly. It's just something [silly]."
232
559000
3000
van dir, "No siguis ridícul. És només una cosa [ridícula]."
09:37
Well actually, I was not silly.
233
562000
3000
Bé, de fet, no va resultar tan ridícul.
09:40
The great painter Hokusai knew it very well.
234
565000
3000
El gran pintor Hokusai ja ho sabia.
09:43
The things on the ground are algae.
235
568000
2000
El que hi ha al terra són algues.
09:45
He did not know the mathematics; it didn't yet exist.
236
570000
3000
No coneixía aquestes matemàtiques; encara no existien.
09:48
And he was Japanese who had no contact with the West.
237
573000
3000
I era un japonès que no havia tingut contactes amb Occident.
09:51
But painting for a long time had a fractal side.
238
576000
3000
Però la seva pintura durant molt de temps va tenir un caire fractal.
09:54
I could speak of that for a long time.
239
579000
2000
Podria parlar d'això molta estona.
09:56
The Eiffel Tower has a fractal aspect.
240
581000
3000
La Torre Eiffel té un aspecte fractal.
09:59
I read the book that Mr. Eiffel wrote about his tower,
241
584000
3000
I vaig llegir el llibre que el Sr. Eiffel va escriure sobre la seva torre.
10:02
and indeed it was astonishing how much he understood.
242
587000
3000
I de fet era sorprenent tot el que hi entenia.
10:05
This is a mess, mess, mess, Brownian loop.
243
590000
3000
Això sí que és un desori total, la corba browniana.
10:08
One day I decided --
244
593000
2000
Un dia vaig decidir que,
10:10
halfway through my career,
245
595000
2000
al mig de la meva carrera,
10:12
I was held by so many things in my work --
246
597000
3000
estava lligat per tantes coses a la feina,
10:15
I decided to test myself.
247
600000
3000
que vaig decidir posar-me a prova.
10:18
Could I just look at something
248
603000
2000
Podia mirar alguna cosa
10:20
which everybody had been looking at for a long time
249
605000
3000
que tothom ja havia estudiat molt de temps
10:23
and find something dramatically new?
250
608000
3000
i trobar alguna cosa completament nova?
10:26
Well, so I looked at these
251
611000
3000
Bé, em vaig fixar en aquestes coses
10:29
things called Brownian motion -- just goes around.
252
614000
3000
anomenades moviment brownià -- només es mou.
10:32
I played with it for a while,
253
617000
2000
Vaig estar jugant-hi per un temps,
10:34
and I made it return to the origin.
254
619000
3000
i vaig tornar a començar.
10:37
Then I was telling my assistant,
255
622000
2000
Vaig dir-li al meu assistent,
10:39
"I don't see anything. Can you paint it?"
256
624000
2000
"No hi veig res. Podries pintar-ho?"
10:41
So he painted it, which means
257
626000
2000
I ho va pintar, cosa que vol dir que
10:43
he put inside everything. He said:
258
628000
2000
va pintar tot l'interior. Em va dir:
10:45
"Well, this thing came out ..." And I said, "Stop! Stop! Stop!
259
630000
3000
"Bé, això és el que ha sortit..." I vaig dir-li, "Atura't! Atura't! Atura't!
10:48
I see; it's an island."
260
633000
3000
Ja ho veig, és una illa."
10:51
And amazing.
261
636000
2000
I va ser sorprenent.
10:53
So Brownian motion, which happens to have
262
638000
2000
El moviment brownià, que té
10:55
a roughness number of two, goes around.
263
640000
3000
un nombre de rugositat 2, ja queda definit.
10:58
I measured it, 1.33.
264
643000
2000
El vaig mesurar, 1,33.
11:00
Again, again, again.
265
645000
2000
Un cop i un altre, i un altre.
11:02
Long measurements, big Brownian motions,
266
647000
2000
Llargues mesures, grans moviments brownians,
11:04
1.33.
267
649000
2000
1,33.
11:06
Mathematical problem: how to prove it?
268
651000
3000
El problema matemàtic: com provar-ho?
11:09
It took my friends 20 years.
269
654000
3000
Els meus amics van trigar 20 anys.
11:12
Three of them were having incomplete proofs.
270
657000
3000
Tres d'ells en tenien proves incomplertes.
11:15
They got together, and together they had the proof.
271
660000
3000
S'hi van trobar, i junts van obtenir-ne la prova.
11:19
So they got the big [Fields] medal in mathematics,
272
664000
3000
I van rebre la gran medalla [Fields] de les matemàtiques,
11:22
one of the three medals that people have received
273
667000
2000
una de les tres medalles que la gent ha rebut
11:24
for proving things which I've seen
274
669000
3000
per demostrar coses que jo ja havia vist
11:27
without being able to prove them.
275
672000
3000
però no havia estat capaç de demostrar.
11:30
Now everybody asks me at one point or another,
276
675000
3000
Ara tothom em pregunta en un moment o altre:
11:33
"How did it all start?
277
678000
2000
"Com va començar tot plegat?
11:35
What got you in that strange business?"
278
680000
3000
Què us va portar cap a aquest camp tant estrany?"
11:38
What got you to be,
279
683000
2000
Què em va portar a ser,
11:40
at the same time, a mechanical engineer,
280
685000
2000
alhora, enginyer mecànic,
11:42
a geographer
281
687000
2000
geògraf
11:44
and a mathematician and so on, a physicist?
282
689000
2000
i matemàtic i encara més: físic?
11:46
Well actually I started, oddly enough,
283
691000
3000
Bé, de fet vaig començar, per estrany que sembli,
11:49
studying stock market prices.
284
694000
2000
estudiant els preus de la borsa de valors.
11:51
And so here
285
696000
2000
I llavors
11:53
I had this theory,
286
698000
3000
tenia aquesta teoria,
11:56
and I wrote books about it --
287
701000
2000
i vaig escriure llibres sobre això,
11:58
financial prices increments.
288
703000
2000
la pujada dels preus financers.
12:00
To the left you see data over a long period.
289
705000
2000
A l'esquerra veieu les dades en un llarg període de temps.
12:02
To the right, on top,
290
707000
2000
A la dreta, a dalt,
12:04
you see a theory which is very, very fashionable.
291
709000
3000
veieu una teoria que està molt, molt de moda.
12:07
It was very easy, and you can write many books very fast about it.
292
712000
3000
Era molt senzilla, i hi pots escriure molts llibres i molt depressa.
12:10
(Laughter)
293
715000
2000
(Rialles)
12:12
There are thousands of books on that.
294
717000
3000
Hi ha milers de llibres sobre això.
12:15
Now compare that with real price increments.
295
720000
3000
Ara compareu això amb els increments reals del preu.
12:18
Where are real price increments?
296
723000
2000
i on són els increments reals del preu?
12:20
Well, these other lines
297
725000
2000
Bé, aquestes altres línies
12:22
include some real price increments
298
727000
2000
inclouen alguns increments reals de preu
12:24
and some forgery which I did.
299
729000
2000
i alguna trampa que vaig fer.
12:26
So the idea there was
300
731000
2000
L'idea aquí era
12:28
that one must be able to -- how do you say? --
301
733000
2000
que un ha de ser capaç de -- com dir-ho? --
12:30
model price variation.
302
735000
3000
modelar la variació de preu.
12:33
And it went really well 50 years ago.
303
738000
3000
I va funcionar prou bé fa uns 50 anys.
12:36
For 50 years, people were sort of pooh-poohing me
304
741000
3000
Durant 50 anys la gent se'n fotia de mi
12:39
because they could do it much, much easier.
305
744000
2000
perquè ho podien fer d'una manera molt més fàcil.
12:41
But I tell you, at this point, people listened to me.
306
746000
3000
Però ja us dic que, ara mateix, la gent m'escolta.
12:44
(Laughter)
307
749000
2000
(Rialles)
12:46
These two curves are averages:
308
751000
2000
Aquestes dues corbes són mitjanes.
12:48
Standard & Poor, the blue one;
309
753000
2000
Standard & Poor, la blava.
12:50
and the red one is Standard & Poor's
310
755000
2000
I la vermella és la de Standard & Poor,
12:52
from which the five biggest discontinuities
311
757000
3000
de la que s'han tret
12:55
are taken out.
312
760000
2000
les cinc discontinuïtats més grans.
12:57
Now discontinuities are a nuisance,
313
762000
2000
Ara les discontinuïtats són una llauna.
12:59
so in many studies of prices,
314
764000
3000
Per això en molts estudis de preus,
13:02
one puts them aside.
315
767000
2000
un les deixa de banda.
13:04
"Well, acts of God.
316
769000
2000
"Be, són imprevistos.
13:06
And you have the little nonsense which is left.
317
771000
3000
I et quedes amb una cosa sense sentit.
13:09
Acts of God." In this picture,
318
774000
3000
Imprevistos. En aquesta imatge
13:12
five acts of God are as important as everything else.
319
777000
3000
cinc imprevistos són tant importants com qualsevol altra cosa.
13:15
In other words,
320
780000
2000
En altres paraules,
13:17
it is not acts of God that we should put aside.
321
782000
2000
no són els imprevistos el que hauríem de treure.
13:19
That is the meat, the problem.
322
784000
3000
Aquest és realment el fons del problema.
13:22
If you master these, you master price,
323
787000
3000
Si els arribes a dominar, domines el preu.
13:25
and if you don't master these, you can master
324
790000
2000
I si no els arribes a dominar, pots dominar
13:27
the little noise as well as you can,
325
792000
2000
el petit soroll tan bé com puguis,
13:29
but it's not important.
326
794000
2000
però no és important.
13:31
Well, here are the curves for it.
327
796000
2000
Bé, aquí hi ha les corbes per a això.
13:33
Now, I get to the final thing, which is the set
328
798000
2000
Ara, ja arribo a la conclusió, que és el conjunt
13:35
of which my name is attached.
329
800000
2000
que porta el meu nom.
13:37
In a way, it's the story of my life.
330
802000
2000
D'alguna manera es tracta de la història de la meva vida.
13:39
My adolescence was spent
331
804000
2000
La meva adolescència va passar
13:41
during the German occupation of France.
332
806000
2000
durant l'ocupació alemanya de França.
13:43
Since I thought that I might
333
808000
3000
I com pensava que podria desaparèixer
13:46
vanish within a day or a week,
334
811000
3000
el dia o la setmana següent,
13:49
I had very big dreams.
335
814000
3000
Tenia grans somnis.
13:52
And after the war,
336
817000
2000
I després de la guerra,
13:54
I saw an uncle again.
337
819000
2000
Em vaig retrobar amb un oncle.
13:56
My uncle was a very prominent mathematician, and he told me,
338
821000
2000
El meu oncle era un matemàtic molt notable i em va dir,
13:58
"Look, there's a problem
339
823000
2000
"Mira, hi ha un problema
14:00
which I could not solve 25 years ago,
340
825000
2000
que no vaig poder resoldre fa 25 anys,
14:02
and which nobody can solve.
341
827000
2000
i que ningú pot resoldre.
14:04
This is a construction of a man named [Gaston] Julia
342
829000
2000
És l'obra d'un home anomenat [Gaston] Julia
14:06
and [Pierre] Fatou.
343
831000
2000
i d'en [Pierre] Fatou.
14:08
If you could
344
833000
2000
Si poguessis trobar
14:10
find something new, anything,
345
835000
2000
alguna cosa nova, qualsevol cosa,
14:12
you will get your career made."
346
837000
2000
ja hauries fet carrera."
14:14
Very simple.
347
839000
2000
Molt senzill.
14:16
So I looked,
348
841000
2000
I hi vaig cercar,
14:18
and like the thousands of people that had tried before,
349
843000
2000
i com els milers de persones que ho van intentar abans que jo,
14:20
I found nothing.
350
845000
3000
no hi vaig trobar res.
14:23
But then the computer came,
351
848000
2000
Però van arribar els ordinadors.
14:25
and I decided to apply the computer,
352
850000
2000
I vaig decidir d'aplicar-hi els ordinadors,
14:27
not to new problems in mathematics --
353
852000
3000
no als nous problemes de les matemàtiques --
14:30
like this wiggle wiggle, that's a new problem --
354
855000
2000
com aquest trencaclosques, això és un problema nou --
14:32
but to old problems.
355
857000
2000
sinó als vells problemes.
14:34
And I went from what's called
356
859000
2000
I vaig anar dels anomenats
14:36
real numbers, which are points on a line,
357
861000
2000
nombres reals, que són punts en una línia,
14:38
to imaginary, complex numbers,
358
863000
2000
als nombres complexos, els imaginaris,
14:40
which are points on a plane,
359
865000
2000
que són punts en un pla,
14:42
which is what one should do there,
360
867000
2000
que és el que un hauria de fer.
14:44
and this shape came out.
361
869000
2000
I va aparèixer aquesta figura.
14:46
This shape is of an extraordinary complication.
362
871000
3000
Aquesta figura és extraordinàriament complicada.
14:49
The equation is hidden there,
363
874000
2000
L'equació és amagada allà,
14:51
z goes into z squared, plus c.
364
876000
3000
z es transforma en z al quadrat més c.
14:54
It's so simple, so dry.
365
879000
2000
És tan senzilla, tan eixuta.
14:56
It's so uninteresting.
366
881000
2000
És tan poc interessant.
14:58
Now you turn the crank once, twice:
367
883000
3000
Però si l'apliques un cop, dos,
15:01
twice,
368
886000
3000
dos,
15:04
marvels come out.
369
889000
2000
apareixen meravelles.
15:06
I mean this comes out.
370
891000
2000
Vull dir que surt això.
15:08
I don't want to explain these things.
371
893000
2000
No vull explicar aquestes coses.
15:10
This comes out. This comes out.
372
895000
2000
Això és el que surt. Això és el que surt.
15:12
Shapes which are of such complication,
373
897000
2000
Figures que són tan complicades,
15:14
such harmony and such beauty.
374
899000
3000
amb tanta harmonia i bellesa.
15:17
This comes out
375
902000
2000
Això surt
15:19
repeatedly, again, again, again.
376
904000
2000
repetidament, un cop i un altre.
15:21
And that was one of my major discoveries,
377
906000
2000
I un dels meus descobriments més importants
15:23
to find that these islands were the same
378
908000
2000
va ser trobar que aquestes illes eren iguals
15:25
as the whole big thing, more or less.
379
910000
2000
que l'illa gran, més o menys.
15:27
And then you get these
380
912000
2000
I tens tota aquesta
15:29
extraordinary baroque decorations all over the place.
381
914000
3000
extraordinària decoració barroca per tot arreu.
15:32
All that from this little formula,
382
917000
3000
Tot això surt d'aquesta petita fórmula,
15:35
which has whatever, five symbols in it.
383
920000
3000
que conté nomès, cinc simbols.
15:38
And then this one.
384
923000
2000
I també aquesta altra.
15:40
The color was added for two reasons.
385
925000
2000
S'hi va afegir el color per dos motius.
15:42
First of all, because these shapes
386
927000
2000
Primer perquè aquestes figures
15:44
are so complicated
387
929000
3000
són tan complicades,
15:47
that one couldn't make any sense of the numbers.
388
932000
3000
que un no trobaria el sentit dels nombres.
15:50
And if you plot them, you must choose some system.
389
935000
3000
I si els dibuixes, has de triar algun sistema.
15:53
And so my principle has been
390
938000
2000
I el meu principi ha estat
15:55
to always present the shapes
391
940000
3000
sempre presentar les figures
15:58
with different colorings
392
943000
2000
amb diferents acoloriments,
16:00
because some colorings emphasize that,
393
945000
2000
perquè alguns acoloriments posen de relleu això,
16:02
and others it is that or that.
394
947000
2000
i d'altres això o allò.
16:04
It's so complicated.
395
949000
2000
És molt complicat.
16:06
(Laughter)
396
951000
2000
(Rialles)
16:08
In 1990, I was in Cambridge, U.K.
397
953000
2000
El 1990, vaig anar a Cambridge, UK
16:10
to receive a prize from the university,
398
955000
3000
a rebre un premi de la universitat.
16:13
and three days later,
399
958000
2000
I tres dies després,
16:15
a pilot was flying over the landscape and found this thing.
400
960000
3000
un pilot volant per sobre els camps es va trobar això.
16:18
So where did this come from?
401
963000
2000
D'on venia això?
16:20
Obviously, from extraterrestrials.
402
965000
2000
Òbviament, dels extraterrestres.
16:22
(Laughter)
403
967000
3000
(Rialles)
16:25
Well, so the newspaper in Cambridge
404
970000
2000
Bé, doncs el diari de Cambridge
16:27
published an article about that "discovery"
405
972000
2000
va publicar un article sobre aquell "descobriment"
16:29
and received the next day
406
974000
2000
i l'endemà va rebre
16:31
5,000 letters from people saying,
407
976000
2000
5.000 cartes de gent dient,
16:33
"But that's simply a Mandelbrot set very big."
408
978000
3000
"Però si nomès és el conjunt de Mandelbrot, però ben gran."
16:37
Well, let me finish.
409
982000
2000
Bé, deixeu-me acabar.
16:39
This shape here just came
410
984000
2000
Aquesta figura va aparèixer
16:41
out of an exercise in pure mathematics.
411
986000
2000
d'un exercici de matemàtica pura.
16:43
Bottomless wonders spring from simple rules,
412
988000
3000
Les meravelles insondables surten de regles ben senzilles,
16:46
which are repeated without end.
413
991000
3000
que es repeteixen sense fi.
16:49
Thank you very much.
414
994000
2000
Moltes gràcies.
16:51
(Applause)
415
996000
11000
(Aplaudiments)
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