Colours Clouds and God_2of4_Clouds are not Spheres
When Nigel Lesmoir-Gordon made the groundbreaking TV documentary, "The Colours of Infinity" about the Mandelbrot Set and fractals - the geometry of roughness, his enthusiasm brought together a dream team of contributors. Sir Arthur C. Clarke presents it. Benoit Mandelbrot, the Belgian mathematician who first coined the term fractal and whose equation, the Mandelbrot Set, would reveal the wonder of fractals only when fed into a computer, explains how it began. Professor Michael Barnsley, the computer graphics researcher who developed fractal image compression technology, explains the applications of the breakthroughs. Professor Ian Stewart, author of "Does God Play Dice?" adds his insights into the beautifully simple equation that gives birth to fractals. A simple mathematical formula has led to a amazing uses in all branches of science, medicine, computer graphics, weather reporting and analysis, geography, topography and even economics.
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TVTranscript
00:00The world that we live in is not naturally smooth-edged and regularly shaped, like the
00:21familiar cones and circles and spheres and straight lines of Euclid's geometry.
00:28The real world is wrinkled, crinkled and irregularly shaped.
00:35It's a wiggly world.
00:37Amazingly, until very recently, we didn't have a geometry to describe the natural world,
00:42but we do now.
00:44It's called fractal geometry.
00:48Fractal geometry was discovered during the 1970s and 80s by Benoit Mandelbrot.
00:55Yesterday was my birthday.
00:57I'm 75 years old, a long life, a complicated life, one that followed very much the pattern
01:04of my work.
01:05My work has consisted in one major good idea, the fractal geometry of nature.
01:12I like the way he started his book, and he says here, why is geometry often described
01:18as cold and dry?
01:21One reason lies in his inability to describe the shape of a cloud, a mountain, a coastline
01:27or a tree.
01:28And so what Mandelbrot has done, he has made some sort of new geometry, which made us able
01:35to describe the nature as we see it around us.
01:38So it made people think in a different way.
01:48Clouds are not spheres, mountains are not cones, coastline are not circles, and bark
01:53is not smooth, nor does lightning travel in a straight line.
01:59When I looked at the book, there was a picture that caught my imagination, that you actually
02:03could describe nature with mathematics in such a good way.
02:08One of the primary goals of science is to find a language that describes the way the
02:13universe works.
02:16Fractal geometry is a new language that has completely revolutionized the way that scientists
02:21look at and explore our world.
02:26Fractal geometry describes the apparently random forms that exist in nature, like rivers,
02:31coastlines, trees, the human body.
02:37I've been concerned with real phenomena, phenomena that nature surrounds us with.
02:45But also with phenomena that man has, in a certain sense, contrived, not quite designed,
02:53like variation of financial prices, like the language of mankind, like the construction
03:00of big computers, in which the whole design is not quite predictable, does not quite follow
03:07from simple rules.
03:09The geometry which I learned in school, the geometry with which our whole civilization
03:16is completely filled, permeated, is that of Euclid.
03:20It is the geometry of spheres, of triangles, of cones, of circles, of straight lines.
03:28But clouds are not spheres.
03:31The shapes of nature around us are very, very complicated.
03:35They are overwhelmingly complicated and they defy description by the tools of Euclid.
03:43To do science, you must describe nature.
03:46Description is a language.
03:47The language can be just words, it can be formulas, it can be shapes, many things.
03:54Fractal geometry has provided the language.
04:06I was born in Warsaw in 1924.
04:09My mother was born in Lithuania.
04:12My father's family was also from Lithuania.
04:14They met in Warsaw, married, had one child before me, had me, and a younger brother.
04:22On my father's side, they were all scholars.
04:24An uncle, a younger brother, my father became a very well-known mathematician in Paris.
04:29He became professor at the Collège de France when I was 13, so I knew that mathematics
04:33was an option as a career all the time.
04:37My uncle had been a very, very important person in my life.
04:41I had another uncle who was my tutor.
04:44That man did not know how to be a tutor.
04:48He was quite amateur.
04:51So what he did was to teach me important things like playing chess, telling me stories from
04:55antiquity, small things like alphabet, table multiplication.
05:02He told me, just go ahead and learn them.
05:04I never did.
05:06Benoit showed an early love of geometry, and he excelled at chess, although he admits that
05:12he didn't think the game through logically, but geometrically.
05:16Maps were another important inspiration.
05:18Benoit's father, Charles, was crazy about maps, and their house was always full of them.
05:24Benoit read avidly and widely.
05:28I didn't go to the first two years of elementary school, then costed later on.
05:34Came to Paris, started two years late because of the change of school system.
05:41I was not really present in class.
05:43I was dreaming and so on without any particular difficulty.
05:48Then the war came.
05:50I moved to a little town in central France called Tulle.
05:53I made many friends in Tulle, in particular one who is still a great friend, who is Pierre Roubinet.
06:03Benoit has always been an outstanding student.
06:06He did brilliantly well at the lycée in Tulle.
06:09He always achieved the highest distinctions in mathematics.
06:13But Benoit has never been completely integrated, certainly not in his youth.
06:19And then later on in life, he never fitted in to the academic institutions.
06:25Then things became very bad.
06:32The Germans occupied southern France, which they had not in 1940.
06:37And I spent a year and a half running around, keeping body and soul together.
06:42In fact, at one point I was an apprentice toolmaker, which I did surprisingly well
06:49because I had a very firm hand and I had a very good feeling of the shape of things.
06:56When he talks about these days and years, it sounds like he always picked up something
07:02fruitful, something positive, something which took him forward another step.
07:09And that seems to be the nature of him, no matter where he is.
07:15Benoit's practical education opened his eyes to forms occurring in nature.
07:20And then, in Lyon for a year of post-high school mathematics, Benoit discovered he had
07:25an extraordinary ability.
07:28I discovered I had a gift of which I was totally unaware.
07:33I could transform any problem that the teacher had contrived to be a very difficult question
07:39of algebra or long calculations, I could transform it spontaneously, instantly into
07:46a question about shapes.
07:47So where he asked for a solution of a certain equation, I would see two shapes intersecting
07:53or two shapes projecting on each other.
07:56In fact, once I saw them as geometric problems, the answer was immediate.
08:01In a certain sense, this revelation, which occurred sort of in January or February of
08:061944, has revealed the principal strength which I had, which has allowed me to go to
08:13many, many different fields, and in each one to transform them into a certain geometric
08:20shape, a certain collection of shapes, which I could live with, understand, become acquainted
08:26with, describe, first intuitively and then mathematically.
08:32At the end of World War II, Benoit returned to Paris for college entrance examinations,
08:37which he passed with distinction and without the usual two years of preparation.
08:43So for about a month, I did nothing but take exams.
08:46To my surprise, it worked.
08:50That is, I could, in a way, cheat, quite legally.
08:54I broke no law, no written law, because instead of doing these very complicated calculations,
08:59I could just see what was happening, really know what was happening, and describe it,
09:04as opposed to search for it.
09:08Benoit won a place to study mathematics at the École Normale, which trained university
09:13and high school professors.
09:16But mathematics at the École Normale was dominated by the Bourbaki movement.
09:21Benoit's uncle Zolem was one of the founders.
09:24Bourbaki's aim was to purify mathematics by making it more formal.
09:29Benoit had problems with this.
09:33The kind of mathematics which I liked was very visual.
09:36The kind of mathematics I liked was very much very close to nature, to hard things.
09:43And that's who was practising the precepts of Plato in most strong fashion.
09:49Plato wanted mathematics to be separate from matter and from sensations.
09:56What I liked was precisely the intimate links between mathematics, matter, and sensation.
10:04I spent a day agonising.
10:06Next day I went there and resigned.
10:07That was a decision which very much controlled my life.
10:11That is, I was in a way accepted into the most exclusive school which had guaranteed future.
10:18The alternative which I took was to go to the other school, École Polytechnique, which
10:23has a much less cohesive plan because it led to a great variety of different professions.
10:31And that really attracted me because I was not seeking a place I would belong very strongly,
10:38but a place which would give me enough freedom, enough scope, enough variety of possibilities
10:44for me to create something that would fit my skills and my personality.
10:50Benoit couldn't bear the triumphant airs and the authoritarianism of that school.
10:56He didn't want to be part of that school.
10:59And for this reason he left the École Normale Supérieure and went to the Polytechnique,
11:04which was much more directed towards the sciences.
11:09He was left alone there.
11:11They left him to think as he pleased.
11:14Bon.
11:15What I wanted to do, as explained to my uncle and to others, was something totally absurd
11:20because it was so romantic and so childish and I was no longer a child.
11:25I wanted to feel the excitement of being the first in a field to bring this element of
11:31rationality, of mathematical structure, a field which may be important or unimportant.
11:39I didn't have the ambition of finding something very important to do,
11:44to have this excitement of finding some field in which I'll be the first.
11:49When I was at the École Polytechnique, a man, the man who had most influence on me
11:53was a professor of mathematics named Paul Lévy.
11:56Paul Lévy was viewed by many as being a very brilliant man, but he was also an oddball.
12:02He was very much a loner.
12:04I was not his student, but he was very influential on me.
12:07He had access to Paul Lévy's work and I think he had a very deep understanding of that work
12:13and working understanding of that work.
12:16And with this understanding, he went into the sciences, into the various sciences,
12:22attacking various problems.
12:24And while doing so, I think he slowly but surely started understanding how
12:31applicable this sort of mathematics was.
12:34After the École Polytechnique, Benoit studied turbulence at the California Institute of Technology.
12:41Then he returned to France to join the PhD program at the University of Paris.
12:46But in what subject?
12:48An odd linguistic observation, Zipf's Law, was brought to his attention.
12:54I was visiting my uncle and asked him for something to read in the subway.
13:00And he reached in his wastebasket, got out a reprint and said,
13:05well, somebody sent me a paper, which is crazy, but you like crazy things.
13:09Here it is.
13:11So I took the subway and the paper was a review of a book.
13:15And the reviewer said that the book describes some remarkable property,
13:20which is true of every language.
13:22French, English, Latin, whatever.
13:24And that stimulated me in a most virulent fashion, that I must find the explanation of that.
13:32After his thesis, Benoit worked for Philips on colour television
13:36and on information theory at the Massachusetts Institute of Technology.
13:40Then he was sponsored by John von Neumann at the Institute for Advanced Study at Princeton,
13:45where he came across the idea of the Hausdorff-Besicovich dimension,
13:50the revelation that there were phenomena that existed outside of one-dimensional space,
13:56but in somewhat less than two dimensions.
13:58Benoit adopted the Hausdorff-Besicovich dimension on the spot.
14:02It was an all-purpose tool and a special example
14:06of the wider notion of fractal dimension, which was to come.
14:10I went to America for a postdoc.
14:11I went to work with von Neumann, who was best known at the time as being the man
14:16who was creating the first very large successful computer.
14:22But I didn't go for von Neumann for the computer, but in general,
14:24because von Neumann was, to me, the example of something that was possible.
14:30Benoit's interest in computers was immediate.
14:33From then on, his use of this new tool grew rapidly.
14:37He returned to France, married Aliette Cagan,
14:41and became a professor at the University of Lille.
14:44His academic future looked assured.
14:46But again, he felt uncomfortable in that environment.
14:50And in 1958, he spent the summer at IBM as a faculty visitor.
14:55It was not a matter of career.
14:57I had a perfectly nice job, but of environment.
15:00Was IBM better?
15:03I didn't know it early on.
15:04I was invited for a summer.
15:06And then in the summer, I realised that actually that place might be the right place.
15:10Not because it was organised to support what I wanted to do,
15:14because it had no idea what I wanted to do.
15:16It was probably better for him to be associated with IBM and give him more freedom.
15:21And that was probably true for me too.
15:23Our personalities don't seem to be well adapted to academic life.
15:30And so it worked out quite well, as long as the good times lasted.
15:35IBM was built primarily by a man named Thomas J. Watson,
15:39later senior, who built a company with building mechanical devices,
15:43sorters.
15:44There was very little modern technology involved.
15:47It was mostly a company based upon very good manufacturing and selling.
15:52And that was coming to an end.
15:54And Thomas Watson Jr recognised that.
15:57He built an entirely different company,
16:00which was going to do electronics as opposed to mechanical engineering,
16:03and based on computers, which were just beginning to exist as a commercial possibility.
16:10Many people came, many people left, a period of great, great turmoil.
16:13But essentially, IBM accepted a dozen, maybe a score, maybe a few more,
16:19of people who truly did not feel comfortable in academic environment.
16:25My role has been, in some sense, a helper.
16:29Very early, when I met Benoit, I realised that he was not quite your ordinary person.
16:36And I think I was among the very first to appreciate the depth and scope
16:43of what he was doing.
16:45Being at IBM, I think, was extremely productive on two levels.
16:50One is this sort of nurturing environment, in some sense.
16:55The other one is the ability to have the computational resources at his hand
17:01before the PC was available around,
17:04and before anybody else actually used a computer as a laboratory.
17:13After the summer was going on,
17:15I decided that coming back to my position in France would be just ridiculous.
17:20I would get myself into some kind of dead end and decide to stay at IBM.
17:25First, it was just because it seemed to be better.
17:28Well, the gamble was very, very successful,
17:31because within about three years, I had made my first truly major discovery.
17:38And that discovery came in a field about which I knew nothing,
17:42namely what was then called economics and later, more precisely, called finance.
17:48How did I come to the discovery?
17:50Well, because of a picture.
17:53I was visiting Harvard to give a seminar in economics on some work of mine
17:58and saw on the blackboard of my host a certain drawing.
18:04I told him, how come you have on your blackboard a drawing I'm going to use in my lecture?
18:08He said, I had no idea what your lecture was going to be about.
18:11This drawing is about the behavior of cotton prices.
18:15Well, we had a chat before my lecture and also afterwards,
18:19and I decided right on the spot that cotton prices deserved extremely careful attention.
18:29And back to IBM, made a certain number of tests.
18:32The tests were absolutely phenomenally successful.
18:37That is, I had, in a very short time, a model of variation of some speculative prices.
18:44In economics, I argued, there's no reason why prices should be continuous.
18:49If a piece of news arrives, the price can go from, say, 100 to 10 or 2 instantly,
18:57or to 300 instantly.
19:00Therefore, I had to accept the notion of discontinuity in the model of proper price.
19:06Now, that idea didn't occur to anybody else.
19:10The beginning of the 1960s, he worked on stock price records,
19:15and that is a very, very influential work, in my opinion, and a very, very good work.
19:21It's incredible.
19:22After 30 years, I suddenly went and read that paper,
19:29and it is incredible to see how much was there already,
19:33although the word fractal was not there.
19:35Everything is described in there.
19:37For example, very beautifully, where he describes that if you look at a stock record,
19:42and if it is a monthly record, a weekly record, or a daily record,
19:47you are not able to see the difference.
19:49And not only did he at that time mention that the thing is self-similar,
19:54but he went all the way to even make models for that self-similarity
19:59and explain where that self-similarity comes from.
20:06IBM asked Benoit to work on eliminating the apparently random noise
20:11in signal transmission between computer terminals.
20:15Now, the errors were not, in fact, entirely random.
20:17They tended to come in bunches.
20:20And Benoit observed that the degree of bunching remained the same,
20:24whether he plotted his observations by the month, by the week, or by the day.
20:29My first work after the prices happened to be precisely a very different kind
20:34about errors in telephone channels, which was something very, very much
20:39what IBM was worrying about.
20:41Because until then, the computers were big boxes in a room.
20:45You went there with some data, went back with some data,
20:48and then the idea of terminals began, therefore,
20:51transmitting data over telephone lines, ordinary telephone lines.
20:56And it was done in a fashion which people felt was very conservative.
21:01But in fact, we were reading very, very complicated and strange results.
21:05And it was a very interesting case in which,
21:08to understand this messiness in telephone lines,
21:13I had used intellectual tools which are the same as the ones I'd used
21:17to look at messiness in financial prices.
21:20And those tools happened to be ones which I'd learned as a student
21:27and which had been introduced into mathematics 100 years before
21:31as being so-called pathologies.
21:34That technology was immediately abandoned.
21:37It was also abandoned by Bell Laboratories and AT&T,
21:40where I gave a talk about it, and they also abandoned it.
21:44A different technology was introduced, which was appropriate to it.
21:48At that time, the objects like Cantor's set,
21:53Peano curve, were known by a few, mostly mathematicians.
21:58And they were convinced that they proved that pure mathematical thought
22:03is quite separate from reality,
22:04because those ideas had no implementation in nature.
22:09I found that, quite the contrary,
22:11those tools of the separation from mathematics and physics
22:17were precisely the tools I needed to understand
22:20this holy mess encountered in those various areas.
22:26These tools, these pathologies,
22:29were strange mathematical discoveries
22:31made in the late 19th and early 20th centuries
22:34by four mathematicians in particular,
22:37Cantor, Peano, van Gogh and Sierpinski.
22:42We know there's a lot of what we call fractals
22:45in mathematics at the turn of last century.
22:49But all of these things like the Koch curve,
22:51like the Cantor set, like the Sierpinski triangle,
22:55were mathematical topics or mathematical shapes
22:59which were not seen in relation with nature at all.
23:03They were seen as examples of a particular
23:07and very peculiar mathematical property
23:10for which they were made at the time.
23:13And they were certainly not made to somehow explore the nature as such.
23:19They were made to explore the nature of mathematical questions
23:24very far removed from nature.
23:27But that was mathematics, you know,
23:29something which was not related to the world.
23:33And Benoist made this connection.
23:37And this is a very difficult step, you know,
23:42this is a very difficult step to make.
23:45Only a few people, in fact, say, well, that's it.
23:49In fact, these things are not important
23:51because of the framework in which they were invented or studied.
24:00But they are important at large, you know.
24:03He saw that. He's a visionary.
24:05The objects existed before as some kind of a paradoxical
24:09intellectual construction that was highly, highly artificial
24:15and therefore wasn't taken seriously.
24:17And a few mathematicians who knew it, you know,
24:20dabbled around and then forgot about it.
24:23But he picked it up and saw immediately
24:25that nature for some reason has chosen this particular feature
24:30as a worthwhile technical help
24:34in accomplishing whatever nature tries to accomplish.
24:37And Benoist saw that very clearly.
24:40I think it took quite a leap.
24:44And that's the big leap which Benoist made.
24:47It takes quite a leap to see that these forms
24:51have something at all in common with nature.
24:54And, of course, now through the work of Benoist,
24:56we know they have more than just something in common with nature.
25:00They are really at the heart of understanding the patterns,
25:06the structure, the irregular shapes of nature in a very deep way.
25:13Cantor's quest for the meaning of continuity led him in 1883
25:18to the set that is now named after him,
25:20one of the first fractals to be studied mathematically.
25:23Take a line, remove the middle third, leaving two equal lines.
25:30Likewise, remove the middle thirds from each of these two lines,
25:34repeat this process an infinite number of times,
25:38and you're left with the Cantor set.
25:40Around 1890, Giuseppe Piano discovered what was called a space-filling curve.
25:47Piano had constructed an idealized curve
25:51which twisted in such a complex way
25:54that it visited every point in the entire plane.
25:58These shapes are incredibly complicated to describe in Euclidean terms,
26:04displaying an endless series of motifs within motifs
26:08repeated at all scales,
26:10like the snowflake curve devised by Helge van Gogh in 1904.
26:15The finished curve is infinitely long,
26:18despite being contained in a finite area.
26:22What van Gogh didn't realise, but Mandelbrot did,
26:26was that such curves with infinite length
26:28would make ideal models for the shapes of the real world,
26:32like coastlines, estuaries and arteries.
26:39The Polish mathematician Waclaw Sierpinski
26:42introduced his fractal in 1916,
26:45but the underlying principles were known to artists for millennia.
26:50The Sierpinski gasket is a shape composed of three copies of itself,
26:55each half as big as the whole.
26:59Now, these shapes all display an endless series of motifs
27:04repeated at all scales.
27:08What is so interesting about Benoit
27:10is that he somehow has this incredible,
27:15how would you say, broad knowledge in mathematics
27:18which goes back into history,
27:20and he seems to have known it all
27:23when he became aware of how these different strands,
27:28how these different sources were playing together in a new way.
27:32So, that makes him unique.
27:39During the 60s,
27:41Benoit's quest led him to study galaxy clusters,
27:44applying his ideas on scaling to the structure of the universe itself.
27:49He scoured forgotten and obscure journals,
27:52and he found the clue in the work of a mathematician and meteorologist
27:56named Louis Fry Richardson, on another piece of waste paper.
28:00Mandelbrot had struck a rich seam, and he knew it.
28:04A very fortunate thing came up, again, from a wastebasket,
28:09more precisely from a pile of material which was being discarded.
28:13The library at IBM was discarding a gigantic amount of books
28:21that nobody had ever looked at because there was no room for them.
28:25But then there was a periodical, which I opened more or less at random,
28:29which I opened more or less at random,
28:31saw a name, Richardson.
28:34And Louis Fry Richardson was a very well-known name to me
28:38because of turbulence.
28:39He was a great hero, a study of turbulence in 1920s.
28:44He was a very strange character.
28:46He was a very great man in many ways.
28:49I mean, a true English eccentric of the years
28:54where such eccentrics were numerous and quite extreme.
28:59So what he had done is to measure the length of coastlines
29:02at different scales.
29:04In fact, I took a Xerox copy of that picture,
29:08then let the book down to pick it the next day,
29:11the book was gone.
29:12I had only this picture.
29:14But that picture was the key to fractal geometry
29:18because it referred to something
29:20everyone knows a little bit about, coastlines.
29:23Richardson loved asking questions others considered worthless.
29:28One of his papers, Does the Wind Possess a Velocity?
29:31anticipated later work by Edward Lorenz
29:34and other founders of chaos theory.
29:37One of Richardson's great insights
29:39was a model of turbulence as a collection of ever smaller eddies.
29:44Mandelbrot was struck by Richardson's 1961 observations
29:48on the lengths of coastlines
29:50and published a paper called
29:51How Long is the Coast of Britain?
29:54This apparently simple question of geography
29:57reveals on close inspection
29:59some of the essential features of fractal geometry.
30:03And everyone knows very well
30:04that if you take a measure of the coast of Britain
30:06at, say, on a sphere,
30:09it's very, very roughened and weighs just a bump
30:13and count, well, maybe a bare bump.
30:15The closer you come, the longer it becomes.
30:17You just look at the coastline,
30:19you see that it's a very shallow strait.
30:23So more or less, you must expect to have
30:25a length that increases as you come closer.
30:28And that's where the dimension,
30:31the fractional dimension, came in.
30:35This article on the coast of Britain
30:36started me quite seriously on the enterprise of fractals,
30:42which went on for many years,
30:44which went in the same fields as before,
30:48in the direction of better understanding
30:51of financial prices,
30:53direction of better understanding
30:55of fluctuations of dissipation in turbulence,
30:59in direction of better understanding
31:01distribution of galaxies in the sky.
31:04In all these areas, and many others,
31:07my procedure started the same.
31:10The beginning was always the same.
31:11I know how to deal with various kinds of messes.
31:16At IBM in 1973,
31:18Benoit developed an algorithm
31:20using a very basic makeshift computer,
31:23a typewriter with a minute memory,
31:25to generate pictures which imitated natural landforms.
31:30Computer graphics became possible.
31:33That was, for me, an extraordinary event.
31:36What do I mean by it becoming possible?
31:38The only place that had computer graphics of proper nature
31:42was, I think, Los Alamos
31:43and perhaps some other military research establishment.
31:47And I was not part of them,
31:48I had no access to their products.
31:51IBM certainly was not making computer graphics.
31:54It was very far from what IBM was good at.
31:58But we made a makeshift computer graphics setup.
32:04First of all, it was a matter of typewriters and getting...
32:09I computed a fake coastline
32:12to fit my theory
32:14and to satisfy Richardson's empirical results.
32:19And this had to sit in front of a typewriter
32:25and eventually the typewriter would type the outline of it.
32:29Now, to tell you how horrible, how hard this thing was,
32:32which is unimaginable now...
32:43I never had the impression
32:44that I was finding the key of everything in the universe.
32:48But the key which I was developing
32:51opened very many, very different doors
32:55to mine and everybody else's surprise.
32:58This became focused in the winter of 72, 73.
33:06I had been away for quite a while
33:08and then came back for a kind of sabbatical in Paris.
33:13I gave several lectures,
33:16including one at the Collège de France.
33:18This was one of the harshest examinations I took in my life
33:24because I was returning to Paris after 15 years,
33:27more or less, of absence.
33:29People had heard my work in finance,
33:32my work on these mountains and so on.
33:35It looked totally incoherent
33:37and they were very curious about my lecture.
33:39I never had the experience of a lecture
33:42forcing me to such feats of reorganization.
33:47And fractals were, in a certain sense, born for this lecture.
33:51The topic I was studying had no name
33:54because, in a way, I was simply...
33:58I was the one to conceive that such a subject,
34:00a study of irregularity, broken openness,
34:04was possible, not on the level of general impressions,
34:09but on the level of very, very sharp, close mathematics.
34:14Benoist published his first book in 1975 in France.
34:20The ideas behind fractals,
34:22self-similarity and iteration, are ancient,
34:25but it took this wanderer by choice to give it a name.
34:29Once named, the field exploded.
34:33As Benoist himself so aptly puts it,
34:36to have a name is to be.
34:40When the book was fairly close to being ready,
34:43I was challenged to give a name to my creation
34:46and then I coined this word, fractal.
34:48Why did I choose it?
34:50Because I had had a classical education,
34:54I had Latin,
34:56and I wanted something which represented the notion of,
34:59as I say again,
35:01all the time broken up and irregular, interrupted.
35:06The Latin word fractus is precisely the proper one for it.
35:12The creation of a word is extremely important, you know.
35:17There are not so many words,
35:18so if you are able to add one word, it's a big thing.
35:22It's a very significant word.
35:24The fact of having a name for it had an extraordinary effect.
35:29It was no longer the crazy collection of Mandelbrot's miscellanea,
35:35but was a study of a certain object.
35:38A name was a form of existence
35:41and that name started early in 1975.
35:47In 1980, Benoist re-examined the maths of the Julia sets on a computer
35:53and discovered the Mandelbrot set.
35:56Ironically, it was Benoist's uncle Zolom
35:58who had directed him to the work of Gaston Julliard and Pierre Fatou
36:03on self-similarity and iterated functions.
36:06So my uncle was always saying it's a beautiful theory,
36:10but nobody had done anything for 30 years.
36:12That was in 1947.
36:13I couldn't see anything to do.
36:15It was one of those perfect egg shapes
36:18and everything that they could think of had been thought of.
36:21But they did not have the computer.
36:26So again, 30 years later, I came back to the same problem.
36:29There existed something people called Julia sets before,
36:33but since people really couldn't plot them out,
36:36except enormous amount of work,
36:38they were really never looked at.
36:40So Mandelbrot was lucky in that he came in
36:44when people started showing pictures on computers.
36:48The computer can do pictures of a complication
36:51well beyond what the hand can do.
36:55What the mind can only imagine,
36:57and I was very good at imagining them,
36:59I could actually see what they were
37:01and look at them and modify them.
37:03I find myself then very strongly feeling
37:07that I was like the 18th century natural philosopher.
37:10And then when the computer came along,
37:12he realized that he could actually draw pictures of these things.
37:16When he saw them, he's a very visual thinker,
37:18when he saw them, the floodgates opened.
37:21Millions of things came out, and that was the beginning.
37:24The year 79-80 was a very important one in my life,
37:29one of the high points I gained.
37:30I had an assistant who was very good,
37:33and I asked him to make demonstrations in the class
37:36about using computer in mathematics.
37:39The students were totally flabbergasted.
37:41It was something astonishing.
37:43They couldn't believe it, that one could look at pictures
37:46and understand many things in mathematics
37:49which they thought were purely theoretical.
37:51But more important yet was the influence of my colleagues.
37:55First of all, they were quite skeptical.
37:58Oh, well, nice pictures, nothing.
38:01Then when I was coming up and discovering more and more
38:04about the structure of what became known later
38:07as the Mandelbrot set,
38:08they were simply stopping me on the horn and saying,
38:11Monsieur, these pictures, what do you find?
38:13Because an immense, fantastic wall,
38:18like a wall of China, was just stumbling down into pieces.
38:23Mathematicians had convinced themselves
38:25of this separation between the eye and the mind,
38:30the idea that mathematicians could never any longer
38:34gain anything from seeing.
38:37This was long past the time
38:40where pictures could help mathematics.
38:42And they were seeing before, in front of their eyes,
38:45new discoveries being made by the application
38:49of very patient analysis of many, many, many pictures.
39:16Well, there are many things which the Mandelbrot set implies,
39:19but I would say as a phenomenon which strikes most of us
39:22as the most unusual about it is the fact
39:26that here you have a formula which is as short and simple
39:30as a mathematical formula can get,
39:32like x squared plus a constant c,
39:37which fully describes the Mandelbrot set.
39:40So you have this very simplistic, basic formula
39:43on one side, and on the other hand, you have an object
39:46which, as you visualise it, you observe it's a kind of infinity
39:52which goes much farther than most of us can imagine something can go.
40:08The Mandelbrot set is the most famous fractal of all time.
40:12It's become an icon of modern mathematics.
40:14It reminds us of shapes that we see in the natural world.
40:19Some have called it the thumbprint of God.
40:23The Mandelbrot set is one of the few discoveries
40:26of modern mathematics to be assimilated by society.
40:42Benoit published his revised and expanded version
40:45of the fractal geometry of nature in 1982.
40:50This book of 82, I think, changed large numbers of people's view
40:55of what science can be and how one can combine
41:01very, very difficult advanced mathematics
41:04with very, very difficult, very low problems
41:09about messy phenomena of all kinds.
41:11I think something that everybody that read
41:14fractal geometry of nature will say is that after reading that book,
41:18you don't see the clouds anymore as you saw them before.
41:21You don't see the trees anymore as you saw them before.
41:23Many, many things change.
41:25Many people reading my book say that could not be true.
41:28He's just making it up.
41:30And so the set-up proved me wrong.
41:34And immense literature came out of people who tried to prove me wrong
41:38and who eventually proved me right.
41:41Much more right than I ever had any hope of being.
41:46Simply, it was the right idea to have at the right time
41:49and I was in the right place.
41:51Fractals are now being used in work with marine organisms,
42:12earthquake data, percolation and aggregation in oil research,
42:17with the formation of lightning.
42:21There are many fractal structures in nature
42:23and what physicists have been doing since the beginning of the 1980s
42:27was to go and try to understand what is the origin
42:31of these fractal structures, the physical origin.
42:34That means what are the physical mechanisms
42:38behind the formation of the fractal structures.
42:41And one fractal structure was lightning.
42:45Lightning resembles the diffusion patterns left by water
42:49as it permeates soft rock like sandstone.
42:52Computer simulations of this effect look exactly like the real thing.
43:01What Luciano Pietoneiro did, that was my PhD thesis advisor,
43:05he formulated a model for the fractal nature of lightning.
43:11So how does lightning come about and become fractal?
43:14And I worked with him on that model for four years,
43:18trying to understand it, doing simulations on that model.
43:21And at the end of these four years, Benoit was starting to get an interest
43:27in this problem of diffusion-limited aggregation and lightning.
43:31And that is when I joined with him at Yale
43:35and we started to work on this problem of diffusion-limited aggregation.
43:43Fractals hold promise for building better roads,
43:45for video compression,
43:48even for designing ships that are less likely to capsize.
43:51Fractal geometry is already successfully applied in medical imaging.
43:56Now, if you start studying the geometry of the particle vein in the liver,
44:02you see that the particle vein is a fractal tree
44:07that subdivides the liver in various segments.
44:12Each segment is irrigated by one particular main branch of the particle vein.
44:19And the interesting thing is that one of these branches irrigates a particular domain
44:26that has almost smooth boundaries within the liver.
44:30So the idea is that if there is a tumor inside the liver,
44:34you would want to identify in which domain of irrigation of which branch
44:40this particular tumor lies,
44:42so that when you perform surgery, you take away one such segment.
44:46That means it is as if you have a tree outside there,
44:49you cut one branch,
44:51and if you cut one branch, you don't damage the other branches,
44:55the other parts of the tree.
44:57So now we have a method that when a patient comes in,
45:00we get the CT scan of the patient with the incomplete particle tree.
45:05We use the methods that are based on various aspects of fractal geometry
45:11to compute from this pruned version of the particle vein
45:16a complete version which gives one the boundaries of the various segments in the liver.
45:24Somehow, fractals seem to correspond to something in our mental set-up
45:32which people find friendly.
45:37Now this friendliness is a great mystery,
45:41but it's a mystery which in a way improves the quality of life.
45:49But it's a mystery which in a way improves upon deepening.
45:56Here is a temple in India, the design is certainly fractal.
46:00In fact, those volunteers have combed the history of architecture
46:07and painting also, but mostly architecture,
46:10and found fractal designs all over.
46:12Fractal geometry did not begin with me, that's for sure.
46:16I organised it, I conceived of it, I developed it,
46:21but some ideas I knew were at least 100 years old.
46:26But I was completely wrong about the date,
46:29because actually the ideas were millennia old.
46:32They are permeating decoration of the Egyptian temples,
46:37of Persian temples, of Hindu temples.
46:47Two very great composers, Charles Warren in the United States and George Ligeti in Europe,
47:05both came towards me with the same story.
47:08They both said that even though very highly trained classical composers
47:16were well aware of all the techniques of composition,
47:21they had always the impression that something was missing
47:23in their understanding of what music was.
47:25I had, as a result of my encounter with fractal geometry,
47:31a deepening sense of the significance of organisation
47:36as opposed to the objects which are organised.
47:40In a very real sense, one can't really make a distinction.
47:42It's like the distinction between form and content,
47:44which is basically nonsense.
47:47A form is not a bin into which one throws a content, certainly not.
47:51But still, if one has all these notes to push around to various places to make a composition,
47:58one can often think of them as objects
48:00and often come to believe in their individual special properties.
48:08Yet a more fractally oriented view, I think, would look at them,
48:13or we should say, in this case, listen to them,
48:15not as objects but as locations in a kind of space-time.
48:33The most fascinating aspect of Benoist is his eternal curiosity
48:40and his willingness to look at things that everyone knows are well understood.
48:46Whenever he does this, he discovers what was well understood is either wrong
48:54or is misleading or is only part of the story.
48:59If you go to fractal conferences, already from the beginning,
49:02they were always multidisciplinary.
49:04The physicists doing fractal geometry were working in chemistry.
49:08They were working in geology.
49:10They were working in parts of mathematics.
49:12So if you would go to a conference like that, you would be learning a lot of science.
49:17At least personally, I think the biggest contribution has been that
49:19this is just a marvelous way to introduce students to science.
49:23It's not extremely large like galaxies or extremely small like atoms.
49:27It happens right where they live.
49:28But it's a language that you can teach people in an hour.
49:33On any number of occasions in the past 30 years,
49:37Benoist has been first obviously wrong and then viewed as well sort of right.
49:45And then people would finally say, well, we knew that all along.
49:48And this has simply happened over and over again.
49:53Fractals are aesthetically pleasing,
49:56often displaying stunning beauty in the most subtle ways.
50:01Even the Mandelbrot set itself defies verbal description.
50:04Fractals are intimately connected with concepts of beauty and elegance.
50:11And yet until very recently, we didn't have a word to describe the familiar shapes of nature.
50:16Now we can see fractals everywhere.
50:27There are people who think Mandelbrot should get the Nobel Prize.
50:29And I agree with that.
50:32But it's hard to find a feel in which you should get it
50:35because there are no Nobel Prize in mathematics.
50:39Long ago as a young man, I had this romantic notion
50:43that what I would like to do best among anything else
50:47was to find some aspects of the world around us
50:51which others would have somehow left alone, not thought of.
50:57This idea was very much mocked, laughed, dismissed by my family and friends.
51:07This search has brought me to many of the most fundamental issues of science
51:14and including some which I improved a little upon
51:19but certainly left very, very wide open and mysterious.
51:23This has been my hope as a young man.
51:26This hope has filled my whole life.
51:29In that sense, I feel extremely fortunate.