[MUSIC PLAYING] LORENZ: Indeed, a pleasure to
introduce to you this afternoon Dr. Benoit Mandelbrot,
who presently is Sterling professor of
mathematical sciences at Yale University and research fellow
emeritus at the IBM Thomas J. Watson Research Center. Professor Mandelbrot is truly
a man from many nations. He was born in Warsaw, Poland
and in the second decade moved to France, where he
subsequently received his doctorate from the
Faculty of Sciences at the University of Paris. And he has spent most of
the latter half of his life in this country. Many people, however,
associate him more closely with another
country, Great Britain, because he asked
in the title of one of his widely-known papers, "How
Long is the Coast of Britain?" Here, he points out
that the coastline is self-similar
under a magnification so that its measured
length will depend upon the length of
the measuring stick. The shorter the stick,
the longer the coast. And, indeed, if it were not for
the incessant waves and tide so an arbitrarily short
stick could be used, the length could go to infinity. His continuing pursuit of the
phenomenon of self-similarity, which he has recognized
not just in coastlines, but in a wide variety
of natural systems, ultimately led him
to coin a word that is now familiar to nearly
everyone, fractals. And today, his
lecture title will be Fractals in Science,
Engineering, and Finance, Roughness and Beauty. His list of honorary degrees
and medals and other honors is far too long for
me to have memorized. I could read it to you,
but this would cut deeply into the speaking time
that is rightfully his. So if you need further
evidence of the stature that he has attained,
simply look around at the crowd that had filled
every seat 10 minutes ago. Without further ado then, let me
introduce Professor Mandelbrot. [APPLAUSE] MANDELBROT: It is, indeed, a
delight to be here among you and, in particular, a delight
to be introduced by Ed Lorenz. We met actually around the
'64 when a mutual friend Erik Mollo-Christensen
told me, well, there is another man in dog house not
quite yours, but a bit further. And so you two lost souls may
find some interests in common. Well, Erik was very,
very, very clear-sighted. Because, indeed, the work
of Professor Lorenz and mine have very many points in common,
even though they never quite [INAUDIBLE] the same field. Now, the title you
see on the screen here is one which,
in a certain sense, summarizes my whole life. I've been writing papers
for about 50 years, which is a fairly long period of time. And as time went on, many
people asked themselves what was the leading
thought which permeated all my
changes of interest from one field to another. And frankly, my answers
were rather cumbersome. It's only lately that I
realized that everything was very simple. All my life I've been working
towards what may perhaps become a sphere of roughness. Now, let me describe this
phenomenon of roughness. If we think of which shapes
in our ordinary evidence are smooth, well, there are
very few-- a plane, perhaps, for a child, a primitive man,
a child once upon a time. A quiet piece of
water without wind. A circle-- well, full moon, the
pupil in the iris of the eye, a few more shapes like that. Straight lines-- very few. On the other side,
the rough shapes are absolutely without number. Wherever we look,
we see shapes which are very, very complicated. Now, the history of
science, as I see today after experience
in many sciences, very definitely continues to
be dominated by its origin. And science began with
our sensations, senses-- the eye, the ear, the feeling
of hot, the feeling of heavy, the feeling of
rough, the feeling of sweet or sour or acid. And each of these senses except
one gave rise in due time, early on actually,
into a science. More precisely, each went
through three stages. For example, in acoustics,
the idea of sound was known from time immemorial
since music always existed. And scales were known, also. But the association of
sound with frequency is the beginning of
acoustics and developed in a very strong fashion. It developed mostly
by a procedure which I'm trying
to describe, which is very important I think. Acoustics did not try
to represent all sounds. That's impossible. It focused on the sound of
idealized strings or pipes and did a marvelous
job with them. For drums, for concert
halls, acoustics is not so terribly perfect. But that is not a criticism. A science does not
have to be perfect. A science must
find in one aspect of the mess of our senses
some substantial part for which a theory can be done. And this theory must
also give insights about the cases to which
it does not strictly apply. And at this point in
time, much of the effort of many scientists is
still, naturally enough, devoted to developing those
senses which I described and also, of course,
to link them. For example--
unified field theory proposed to link the
feelings of heavy and the feeling of
light, light and weight. But in all this
extraordinary effort, one sense has become
totally deprived of implementation in science. And that's the
sense of roughness. The problem goes
very, very far back. Being very much one of those
nuts who read the old books, I was very impressed to find
in Plato, one of his dialogues, a list of the senses
and what I just said, the sciences
corresponding to it. And then he says, roughness is-- well, he didn't
say much about it. And roughness, indeed, turns
out to be far more complicated than the other sciences
to require mathematics which is a markedly higher
level of complication end of difficulty. So what my whole life has aimed
at was to find irregularities, invariances-- since
science is only study of invariances-- invariances
in our experience of roughness, which might apply throughout
and might provide tools for attacking roughness
in all its aspects. So when I say that fractals--
which is the word I coined for a certain kind of
orderly roughness-- they occur in mathematics,
I will give you at least one example. The sciences I will give you
a few examples in passing-- engineering, finance. I'll spend some time in finance. Now, sometimes one wonders
why all these various topics are brought together. They're not brought together
for any fundamental reason except that they all
exemplify, once again, these very simple and
fundamental issue, the question of roughness. So as I see on the
first line, fractals are simple, complex,
and open-ended. What do I mean by that? If one begins to define fractals
without the kind of cloud of sophistication, rigor, and
mathematical elusiveness which was given to them for quite
a while when they are purely part of mathematics,
if one forgets about these complications,
axiom is simple. In fact, children are taught
the ideas very, very early on. But these very simple rules
create extraordinarily complex reality and very rapidly,
as I will show you shortly. And it's open-ended in a sense
that one very rapidly goes to problems of such
extraordinary difficulty that the best minds are
struggling to cover them. Now, everybody knows
me for pictures. And for some, pictures
were quite glamorous. I'm very, very happy about them. I'm delighted that an
instrument like the computer, which at that time
was devoted entirely to arithmetic
operations and the like, had been tamed for the
purpose so different from its original purpose. But my experience is that if I
started showing you pictures, I will never end. Great fun for you for me,
but it's out of place. So I will simply
not dwell on them. I will just show you this one. [LAUGHTER] Well, let's now get to my point. What is roughness
and how to handle it? Well, it's very appropriate
that this auditorium is in a building
of Earth Sciences, because an Earth
scientist, [INAUDIBLE],, gave me this picture. He told me that when he teaches
geology to his students, he's very careful to tell them
at every given time always put them into scale. Because if you don't
put them into scale, you'll forget how
big the thing was after [INAUDIBLE] photographed. So you put yourself, the
camera cap, or something in it. And then you remember
how big it is. Otherwise, you forget. You confuse a pile of
rocks and a mountain. Well, here is an
example, indeed-- [LAUGHTER] --of this confusion. This one's cheating. It's very easy to cheat. Because, indeed,
this type of terrain is the same at all scales. Now, to be the same at all
scales is, at this level, an element of purely folklore. And I'll hasten to go ahead and
speak of finance, since I spent many, many years of my
life both in the '60s and again recently in
studying financial crisis. In finance, there is
also a piece of folklore which says that, if you put
a price graph somewhere, be careful to put
a scale of time. Because, otherwise,
one would not know whether the price
changes or price is every minute, every second-- well, that's recent-- or
every day, or every year. It's all the same, goes up
and down, zigzags and so on. This is folklore. And I'm not at all ashamed
or embarrassed to speak of folklore. Because in a field like optics,
folklore has been, how to say, taken advantage of a
long, long time ago. In the fields which I'm
interested in, in fact very often, very
little was known. Therefore, folklore is
very, very essential, especially if the folklore
can be, how to say, changed, modified to
fit the other purposes. Now, what is this
folklore about primarily? Now, this edible vegetable
indicates its structure. It is called
cauliflower, obviously. Its most striking feature,
as everyone knows, is that one takes a floret as
opposed to whole cauliflower and drops everything
else, the floret looks exactly like
a small cauliflower. And then I can do it
again, again several times. Well, it's very interesting
that this feature was known to everyone, but I found
very little mention of it until my work, perhaps. The cauliflowers are
very interesting. They look not from
side, but from the top. And there are all kinds of
Fibonacci series, all kinds of structures in them. These attracted
very much attention. But I'm not
interested here in how to say precise description of
the arrangement the florets-- on something much,
much more important, which is the roughness. Certainly, surface of this
vegetable is very rough. It's very rough in a
very excessive fashion. That is hierarchical,
the hierarchy, a big thing made of parts
identical to big thing and so on and so on. The parts are well-defined. Now, the world in general
is not that simple. And in general, this idea of
large and small scale parts is, of course, all
mixed up and confused. Now, this marvelous
graphic is due to my friend Richard Voss to implement
a model of mountains that I put forward way
back in the early '70s before computer was able to
do anything of this sort. Now, it is not a
picture, photograph. It is not a painting. It is the implementation
of a mathematical idea. And the mathematical idea
is so abstract and so, how to say, devoid of
substance that I was very much [? attacked ?] on this account. The idea is simply that
there is an invariance between the roughness
of mountains in big, small, and
very small scales. Now, again, it was not
something which I, how to say, quote unquote "invented." Explorers of the mountains-- Whymper, who climbed the
one Matterhorn, at least was the first man recorded
as climbing Matterhorn, wrote about his
rambles in the Alps that a small part
of the landscape is the same as a big part. And he had all kinds of
philosophical consequences from it. But it was certainly
not taken seriously. So what I did here
was just to embody this idea of this
wisdom of folklore and replace by invariance. Now, one knows the
invariance which occur in relativity theory,
quantum mechanics, and so on. Here, the invariances are those
of dilation and reduction. If you dilate it and
reduce it appropriately, it doesn't change. Well, we can go on forever. But I would like,
first of all, to give an idea of the nature
of the problem and just how it's passed. Fractal is a word
I coined in 1975. The idea of fractality
did not exist before, but an intuition of
fractality and, in fact, examples of use of
fractality go back absolutely to time immemorial. This picture here is that
of a village in Tanzania of a nation which I
don't think lives there. It's a ruin. But if one looks at
this village carefully, if one looks mostly
from the top which is less nice photograph
but more telling, one realizes that's
made entirely in hierarchical fashion. There's a whole village here. This thing you see here, which
is the harem of the king, then the king's house and
so on, everything is on the same pattern,
but more or less big. Humanity has always
known of that, but this was decorative device. And the decorative device was
known, again, to every country. I chose Africa. I might have chosen
just as well temples in many parts of the world. Now, we get to a point,
or two points, rather. To the right of this picture,
you see the celebrated Face of War of Dalí. I don't think Dalí
knew about fractals. To the left, you
see this [INAUDIBLE] which has a long history. I first discovered it myself--
"discovered," quote unquote, "invented," quote unquote. And then I realized that
a man named Sierpinski had written a paper about it. And [INAUDIBLE] a
name called Sierpinski gasket to make it sort of nice. So it has become a very
important shape in physics. You see exactly
how it's built. I don't know [INAUDIBLE]
to tell you. You take a triangle. You take away small triangles. Now, the most astounding
fact is that you go to the Sistine Chapel,
you find approximations of it on the pavement. If you go to churches, you
find these kind of structures all the time. It has been all along a
very long decorative design. So to summarize, fractals have,
on the one hand, a very, very old history. And then this
shape, the triangle, was drawn first around 1900. Then there was a period
in which mathematicians started distilling them. Did they know about the uses
of fractals in decoration I don't know. There is no record
of the telling. However, Cantor, who was one
of the greatest among them, did write the record in
writing of his thinking. And he certainly wrote
to his friend [INAUDIBLE] repeatedly that we mathematician
are of a divine race, and we can devise shapes
that nature does not know. So at least some of them thought
they were inventing something. In fact, they were not. They were putting in a
very, very precise form some structures which humanity
had been always familiar with. And so then the question arises,
if you want to begin a science, you certainly must
need numbers for it. And the astounding finding was
the case at least 15 years ago that whereas, of
course, loudness had been measured very well-- a pitch can be
measured very well by frequency and same thing
for visual impressions. Weight has been measured. Temperature had been measured. That was actual invention
in history by Galileo. There was no measurement
of roughness. And I wrote in the mid-'80s an
article with some metallurgist friend about roughness. And he showed me
after many examples that in the literature of broken
metals or fractures of metals, there are all kinds of
measurements of roughness which are very, very imperfect
and very, very unsatisfactory. Well, so I introduced
in science a notion which I call fractal dimension,
which takes many shapes. I'm going to go very
quickly through it. It is a simplification
of something which had been used before, used
only in mathematical esoterica by Hausdorff. And that simplification,
that Hausdorff dimension is impossible in science. There is no way to measure it,
because operations it embodies. The other dimensions
are, indeed, measurable. And in fact, something
very interesting happened. You will see here these
dimensions can be fractions. And after several
years of working with them, in fact doing
nothing else except using them in one context or another,
I didn't [INAUDIBLE] I taught myself about roughness. Because everybody, I
think, knows about that. I taught myself a
correspondence between roughness and this number. An anecdote-- at one time,
a friend came to visit me-- I had then two programmers-- to show us a beautiful
new construction. And he asked me to
guess dimension. My two programmers who
had no experience of it, one said 1.2 and the other said
1.8 just out of their hats. And I said, it's a
little bit short of 1.5. It was 1.48. And then I'll come
back to an example where I guess dimensions
for [? thirds, ?] because of being attuned to it. Again, I don't think that I
have a skill which is unique. I established myself
a correspondence between this number. Now, how this number come? It starts with a very,
very trivial property of dimension in ordinary cases. You divide a string into n
parts or a square handkerchief in n parts. In each case,
dimension is the ratio of log of n divide
by log 1 over r. Whenever I publish that
in scientific journal, an idiot editor tells
me which [? base. ?] [LAUGHTER] Of course, it doesn't
matter, because the ratio. Now, instead of putting
the usual shapes, you take an interval. You replace it by the zigzag,
[INAUDIBLE] four parts. And you repeat again, again. Well, here, if you take
log n over log 1 over r, you get 1.26, et cetera. In the middle part,
you get 1.5 exactly. In the right part, you get 2.0. I'm not going to go
into details of it. It will take forever. However, these
[INAUDIBLE] quantities have been totally
tamed to be measured. They can be measured sometimes
with exquisite precision. In one object to which
my current latest paper is devoted, the
dimension is a 1.71 with some uncertainty
about the next decimal. It's, well, a little
bit uncertain, not much. But it's astounding
that the notion which is so abstract when properly
generalized and so on becomes so completely measurable. And so let me now bring
this matter [INAUDIBLE].. Now, this picture is in my book. I forgot which page. I was very much in
this Brown motion. Brown motion is one of the
center of probability theory. Of course, it's a
process which was defined by Norbert Wiener in '20s and
was his greatest claim to fame for a long time. It's a drunkard's walk. And you see this black
line goes around. So I was drawing these
things all the time. And in a certain
sense, I was fishing. What I mean by fishing? Well, I have very
strong impression that my colleagues and
friends just underestimate the power of their eye. Well, there's
something else to it. People are more or less
good at it with the eye. I mean, it's very clear. Scientists are
[INAUDIBLE] mostly on exams, which measure
ability to do algebra fast and correctly. So it's not necessarily
the case that scientists have very good eye, at least
not in mathematics and physics. Certainly, in other
fields, the answer is yes. But I do have a great
dependence on my eye. And I was playing, and
playing, and playing, and then decided that this
thing was a bit too diffuse. And the Brownian
motion was [INAUDIBLE] here and going around,
around, around, ending here. So everything I was
looking for was invisible. Everything I was looking
at had been known before. So I decided, first
of all, to have the snake bite its tail by
having Brownian motion come back to where it started. And that's, again,
idea of Norbert Wiener. It's called a Brownian bridge. And then I decided to color it. Color doesn't mean I put
fancy colorful things like on these mountains
on the Mandelbrot set. It's just very simply to make
a difference between inside and outside. So all the points which could be
gotten from outside are white. All the points which are not
accessible are put in gray. When I saw this picture
first on screen, I tell you I
[INAUDIBLE] speechless. This picture screamed at me. I am an island-- [LAUGHTER] --a very complicated island. Now, as Ed has said, the "How
Long the Coast Of Britain?--" and you can replace any
island as you wish in it-- is the very basic issue. And the length is
quite irrelevant, because it depends upon
how you measure it. The roughness is
measured by dimension. And so I've seen many islands,
real ones and fake ones. And this one was
very, very irregular. So my first reaction was about
the most difficult islands I've seen were for 4/3. 1.5, it goes beyond. 4/3 was the kind
of magic number. 4/3 was the islands I was
very much pleased with as being sort of irregular. They measured it, because
man can measure it. And the answer was that
the dimension was 1.33336. Well, this is a
humongous simulation, so you can get these
things very accurately. It was 4/3. Now, the most extraordinary
thing about that is that, if you begin
to be attuned to it, you look in these
kind of contours for all kinds of shapes. And some people claim
that the left part of it is Spain, which had been sort
of made bigger, that England, of course, had been vanished
[INAUDIBLE] different island. Scandinavia was there
a bit out of whack. Anyhow, people begin to see
actual geographical features very easily. And that's sort of
almost automatic. You do curves of this
dimension, and you find it. Well, 4/3 was for a long time
a conjecture for 80 years. Then a friend of mine,
[? Bertoli Brontier, ?] gave a demonstration
of the 4/3 which was very beautiful,
but not rigorous. He calls that exact,
but not rigorous. It uses alternate
techniques of physics, which are probably correct,
but are not certified as such. And then shortly afterwards,
Schramm, Lawler, and Werner proved it. Now, what is
interesting here is not that the theorem was proven. Actually, the proof takes
several hundred pages. It's extremely difficult,
techniques of every kind. It's some overwhelming thing. The interesting thing was
the extraordinary short path between a childishly simple
thing, namely a drunkard going around, and then a question
which is very difficult, which for 20 years baffled the most
brilliant among my friends and was topic of great despair. First of all, they
said, I'm going to do it overnight, then next
week, next month, next year. Finally, it took a millennium. Well, it didn't
take a millennium, but it had to wait
until next millennium. It's very often the case. And I think that actually
it is representative of a feature what I say about
roughness, that roughness have not been explored
systematically for itself. I mean, I made a lifetime
devotion to it, but was not. Therefore, it is not a case
in which simple problems have been solved long ago and the
only difficult ones remain. Simple to state, I mean. It is also a field
which is not very old. So there are very few cases
in which a problem arose out of previous problem. Problems, mathematical
questions arise out of things which are
very, very common. And I will tell
you at the end more about my very gay
commitment now to education. And one of the
aspects of education is to realize that even
adolescents, sometimes your small children,
junior high and so on, are very comfortable with
the concepts that describe roughness, the concept
of self-similarity, the concept of algorithm. I mean, if no
obfuscation is added, they understand very well. And then when they see that
by a little manipulation from these very,
very simple things, one gets these very
complicated matters, I think that we make them
understand to at least some extent this notion of
extraordinary fruitfulness of science in creating
problems as it develops. And, well, it's a
course at Yale, which I expressed a very [INAUDIBLE]. I will tell you more about that. It said the last line,
"token of the value of the eye's recent return
into pure mathematics." Expression of that is that-- there's an article about
that in the French edition of Scientific American. And the illustrator
felt obliged to mark the boundary of this object. The whole point--
that the boundary didn't have to be marked. The boundary was visible
without being marked. You didn't have to mark it. Well, all that cries
out for simpler proofs. And I'm going to go further. The critical application
clusters I would not go into details of what it is. But the general idea
is very close to what the structure of
magnet [INAUDIBLE].. A magnet is made of little
magnets going up or down. And our critical percolation
clusters complicate shapes. And they, too, have boundaries
of dimensions, 4/3 and also 7/4, because depending
what you look at, you get either 4/3 or 7/4. It was the object of
great fascination. [INAUDIBLE] measured
after I [INAUDIBLE] 4/3. And, again, [INAUDIBLE]
did something about it. And [? Stan Smirnov ?]
proved it earlier this year. It's a whole field
which has arisen which now is completely
independent of the eye, of course, but which would not
have arisen without the eye. And I really think that it's
very important from a viewpoint of an understanding of science
to realize what has happened. The reason why the eye went
out of the hard sciences was not a, how do you say,
decision by some committee to neglect it. It was observation
that, as time went on, new questions did not
come from the eye. New questions came
from old questions. Also, the fact that once those
sciences were established, again, the simplest things
were very rapidly done. And then the more
complicated things came from simpler things
in the field itself. But in the case of the
prices of roughness and of the topics I've
discussed all my life, situation is very different. It's still very, very close
to the initial basic issues. And therefore, the eye had
been absolutely indispensable, the eye and sometimes
even hearing. In many cases, I have
transformed some signals into noises which are audible
and discovered, in so doing, that phenomena which
previous analysis had declared to be identical. They were, in fact,
very, very different. They sound very different. Now, I would like to just
make one rapid step back to the Mandelbrot
set, which I didn't want to spend all my time
on, in fact almost no time. When I became fascinated
by the subject, discovered it and described
it around '79, '80, I started with a
definition which is different from the one
which everybody now uses. And it turned out that
it's very difficult to do numerically
on the computer at a time, very clumsy. And so I changed definition,
which is a process mathematicians like. But in this case, I
changed definitions thinking the two
definitions were identical. And the second definition
is one everybody knows. It's very easy to program. And all these incredible
riches came out pouring. And observations
made on the pictures became mathematical conjectures. And they're modified,
proven with great success. But the initial idea that the
two definitions are identical, where one set is the closure
of the other, that assumption, conjecture, whatever,
is still not proven. We use that and Brownian
cluster in this course at Yale on fractals for
non-mathematician and also in workshops
for high school teachers, because there is
nothing there which has to be left unexplained. Everything is there will
explain in great detail. And they understand
very well the nature of some problems, which
otherwise would not be understood. Now, let me change topic
substantially, but not so brutally. Because what I'm going to say
applies not only to prices, but also to many
noises, many phenomena in nature in many fields. But this corresponds to one
of my earliest fascinations with financial prices in the
early '60s and then a topic which I very much devote
a very big part of my life to at present. Now, prices, of
course, go up and down. That was known to everybody. And there are all kinds
of nice maxims about it. In 1900, an incredible
genius looked at a problem. His name was Louis Bachelier. Nobody noticed him. He had a very miserable life. But he wrote in 1900 a
PhD thesis in mathematics, believe it or not, called
the Theory of Speculation. Speculation meant speculation
on the stock market or bond market. And he introduced
for the first time, in loose and incomplete
fashion, the Brownian motion which Wiener later made into
a central mathematical topic and which [INAUDIBLE]
Einstein and others made the central physical topic. Now, the idea of
Bachelier was more or less that prices vary at random. You can't predict them. You toss a coin. If it's heads, your price go up. It's tails, price goes down. And you go on and on and on. Well, you can approximate
that by sequence of independent
Gaussian variables. And you see it on
top very clearly. The sequence of independent
Gaussian variables, which are the increments
of Brownian motion, it's called also white
noise, carelessly called white noise,
which we call Gaussian, Gaussian white noise. It makes a very big difference
I am going to argue in a second. Now, much later, a whole
theory of stock market occurred on the basis that
this model Bachelier is indeed a representation of reality. And the size of these
increments of price, this size is what's called volatility. The model assumes
constant volatility. And indeed, if once you've seen
a little piece of this line on top, you have seen them all. They're all very much alike. And if you average it, if you
take averages, moving averages, this whole thing goes to
zero very, very quickly. Now, look immediately at
the bottom five lines. Skip two, I'll come
back to them later. At least one of them is a real
price series, price changes. Or at least one of
them is an illustration of my current forgery, which is
called the multi-fractal model of price variation. Now, I'm sure that
many, many people here are such great geniuses
that you guess immediately which is which, but
you are exceptional. Most people can't do it. Because all these phenomena have
a very strange characteristic. They have very big
peaks all the time. And the peaks don't
arise by themselves. They arise in the middle of
periods of great volatility, then there are periods of
very, very low volatility. There are periods where
volatility suddenly changes. Volatility, as you try to grab
it in these sequences, either the fake or the real ones,
is something very, very, very elusive. In fact, it's
impossible to grasp. And so how are we
going to be represent that [INAUDIBLE] phenomena? And I put this
example in particular, because it puts
in contrast very, very sharply the
approach to roughness which has been used by,
I think, most authors before fractals and the
approach to roughness which I've been advocating. And I would like to
emphasize differences without going into
long calculations. The approach which more
people have been advocating is, if a constant volatility
phenomenon doesn't work, then replace it by something
which is a variable volatility. And if that doesn't
work, then replace it by something at
discontinuities or jumps. Well, this is very
much the spirit of the celebrated model
of the motion of planets due to Ptolemaeus. In the Ptolemaic
model, the planets go around circles and then
there are epicycles, which are-- and then the
epicycles, planet is going on circles, which center
of which goes another circle. And then in addition,
we must assume that the center of
the basic circle is not identical to the
center of the universe. In fact, there
are three factors. Everybody knows how
Kepler replaced that by a different formula which
later Newton transformed by explaining it
through gravitation. But the main fact
is that there are two approaches to
this phenomenon, either that the prices
which are in the lower part are just other
examples of something complicated, unpleasant
which are just combinations of the simple, or that there is
something radically new which is involved. And I'll tell you
about how radically new this thing has to be. Many of these very big peaks
we see on the five lower lines exceed, well, something which is
defined as standard deviation. Actually, I don't know
how to exactly measure it, but sort of 10 times bigger than
the standard deviation of most examples, the quote
"10-sigma events." Now, 10-sigma events in
the Gaussian process, Gaussian independent
process on top, the probability would be
one millionth, a millionth, a millionth, a millionth. The inverse of Avogadro' number,
it's a very small number. If that were the case, these
things will never happen. But as you see here, they
happened all the time. So it's not a matter of adding
a little bit of large values, because one gets in a situation
in which the large values dominate everything
very strongly. As matter of fact, I'm going
to go a bit illogically to make a point early on. The central aspect of
statistical physics, statistical thermodynamics,
is that a big system evolves in a certain way, namely entropy
increases with probability one. With probability zero,
entropy is going to decrease. But the events of probability
zero are totally negligible. The events which count are
those of probability one. Average, of course, converge. The past and the
future are independent. All kinds of beautiful
theorems apply. But if you go to
financial prices, it's, again, part of folklore. If you look at last
10 years and wonder, assuming that somebody
with miraculous powers of seeing in the dark some
devil would pick the days that matter, you realize
that the days that matter were just 10 days
out of last 10 years. Very few days matter. And the great fortunes
were made in very few days. Great ruins happened
very few days. So one gets ultimately
the situation which is very, very unsettling. That is, in this
context, already very few rare events
count overwhelmingly. And the rest doesn't
count hardly at all. It's a context in which one
must review one's intuition very strongly. But reviewing and
reviewing, it's best not to abandon
[INAUDIBLE] and abandon the primary principle
of modeling, which is that of invariance. It becomes almost a joke-- I said to you before-- to others, to what extent some
scientists, I am one of them, emphasizes that basic trick
of science is invariance. You find the right invariance. And in fundamental physics,
you look for all kinds of new invariances. And the lucky ones or
the [INAUDIBLE] ones are the ones who
find better theories. In this case, the matter was
to find the right invariance. And so I'm giving you the key. The forgeries which I
introduced in among these lines were not chosen by
taking the line on top and adding this, adding
that, adding this, adding that until the thing looked
more or less reasonable. It can be done, but I
don't do it this way. The key is to find
the right invariance. And lo and behold, the right
invariance gives this result. Now, let me again
take a step back, because it's a lesson
which not only applies to this, financial prices,
but too many other phenomena. Believe it or not, but
the five bottom lines are all white noises. I repeat. They have a spectrum,
which is constant. There's no correlation. But [INAUDIBLE] it's full of
very interesting structures. How can a white noise
be so interesting? I'm used to white noise like
on top, which is boring, which is like you've seen one
Redwood you've seen them all, like former President
Reagan said. At the bottom,
everything is different. Well, the fact is that the
bottom five lines are not Gaussian. Therefore, if not
Gaussian, whiteness does not imply boredom. Whiteness implies, very
often, a very rough ride. So a tool of science,
namely the spectrum, which had been marvelous is
not applicable in this context. It doesn't work. I mean, again, Wiener was
very clear when he described spectral analysis early on-- as you realize, Wiener was one
of my intellectual heroes-- when he described the condition
which spectra are good. But these conditions that are
not satisfied for these data. Now, is that a new finding
by [INAUDIBLE] Mandelbrot that these are white? By no means. I remember when spectrum became
usable after the Fast Fourier Transform was discovered,
rediscovered-- rediscovered, actually. Everybody went on to spectral
analyze everything in sight. And prices were
spectral analyzed. They turned out to be
white, price changes. And the result was
absolute, absolute horror. How can it be? If it's white, it should
be like line on top. Now, every book of
mathematics says that independence and
orthogonality are the same for Gaussian processes. Orthogonality means a white
spectrum in this case. But nobody paid attention
to it, because the examples given when they are different
are very made up examples. These are not made up examples. They are examples
made by culture in so far as stock market is
not part of nature, therefore it's part of culture, or made
by me which is also cultural. But so you can have
structures in this fashion. Now, the other
thing is the habit came from thermodynamics
that, if you take a big enough system, then
fluctuation does not matter. In fact, the whole thing
was proving larger, bigger and bigger system. Ultimately, fluctuations don't
matter, and they average out. But look at this
thing on the left. It is not quite the same thing. On the right, it's
squares of it. And there's some
hanky-panky to it, but it's about,
roughly speaking, variation of prices
over a century. If these little
things right were just fleeting defects which average
out, then over a century they've been very, very smooth
indeed and certainly not smooth at all. So we did a phenomena in which
averaging does not really make any simpler. And that's, again, the signature
of an invariant processes, the same as small
scale, big scale. Self-similarity or variance
thereof enter into the game. So the models I'm going
to need to say a few more words about that
multi-fractal models. But the second line on top
was my first model of prices. I call it mesofractal. Now, it's a new
term, because I need these terms for various reason. At that time, I was so
focused on large values that I neglected the
dependence between prices. The bottom lines
show you the prices have enormous dependence. I mean, that's why people can-- there's a [INAUDIBLE] of
prediction actually possible-- well, in one form or
another, but that's a different issue here. But the second line
was independence, had these very long peaks. And that took care of
one of the problems which is the long term dependence, the
long tails, the large values. Then I had another way. The second line took care
of long-term dependence. But only multi-fractals
took care of both. It's interesting
that multi-fractals and multi-fractality is a
notion which I introduced about the time I met Ed
Lorenz, a matter of fact, a little bit later to explain
the nature of dissipation of turbulence-- not explain, I'm sorry. I take it back-- describe it. Explain is a big word. Describe, because
it's a topic in which truly the natural tools borrowed
from other successful sciences have been quite unsuccessful. Now, let me explain
to you how here this matter of
self-sameness which is no longer self-similarity,
but self-affinity, entered [INAUDIBLE]. So this is not a
model of price change. It's a cartoon. What's a cartoon? A cartoon is something which
has recognizably same features, but which is simpler
to draw and to behold. So on top, you have
a straight line trend which the price changed
from beginning to end. I was told, by all means,
it must be going up. So it's going up. And then there is this
variation of this price, which is the second line,
which I call generator. It goes up, up, down, up. And to simplify it,
I make it symmetric. Then for each of the
next stages, what I do is we take this
generator and squeeze it. But, you see, you must squeeze
it in different ratios this way and that way. If the ratio of squeezing
were the same both directions, you would deal with similarity. And the [INAUDIBLE]
is self-similar. We squeeze the different
rates this way and this way. It is an affinity
mathematically. Therefore, these
things are self-affine. And if it goes down,
you turn the thing back. You do it again, again. Now, after a while, you
get this jagged curve, which has a certain amount
of boring regularity. So you inject the minimum
amount of randomness into this. So you simply, before
you introduce generator, you choose at random
between two possibilities, down up, up, up, down,
up, or up, up, down, very easy to simulate. And then here is what you get
with different [INAUDIBLE].. On top, with the values you
saw in previous picture, you get the sequence which is
more or less what Bachelier was saying, more or less what
the conventional so-called theory of the market supposes,
coin tossing or a variant thereof. Then what you do is-- to come back to
previous thing-- you move this top point a
little bit to the left. So you have one parameter
here, [INAUDIBLE] one point parameter. The point parameter is the first
break determines the generator. And so as you do it, you
realize that suddenly this thing becomes more and more irregular. Now, this is all there is to it. It is not an
incredible accumulation of cycles, epicycles,
not-centered circles, et cetera, et cetera. It simply is an affinity. These curves are increments of
a self-affined process which is self-affined by
construction, because you do it by recursive process which
has the minimum amount of randomness, namely shuffling. And at the end,
you get something which is no longer
realistic for stock market, because it seldom happens that
you have these periods where almost nothing happens. If you went further,
you get even further. So you can go beyond what
you observe in stock market. Now, I don't say it's a model. I just say that it's a case
in which the first investment gives a high yield. And I have been
in many sciences. And I'm also history nut, so I
read records of many early days or many successful
theories in science. And there is one theme which
runs through them, which is that the first stage
was awfully simple and it gave interesting results. Then, well, bells and whistles
were added, but only gradually. And that those
theories, which began by design which was full
of bells and whistles to make it look
right, never take off. They are just too
overburdened to begin with. Now, I don't say this
one is going to take off. It's very widely respected. It's very widely criticized. It's very widely hated. It's very widely et
cetera, et cetera. But the point is
that it provides what I think was
necessary in this context, to have an extremely short
no bells, no whistles path between folklore and a
quantitative measurable and also realistic
representation of data. But, now, here is something
which is very disquieting. So you see on top of these
things, this is zigzag? It's not the same as before. But, again, you
just take zigzag, and you do it
recursively by changing the sequence each time. And so these eight
sequences or increments are identical from
viewpoint [INAUDIBLE] process generates them. And here we get into one of
the most constant and the most, in fact, perturbing features
of fractal phenomena. And perhaps I would first-- or, no. Let me just comment on that. If you apply to these eight
processes, the usual techniques which are applied to
randomly varying phenomena, you find that they're
all different. The variance is different. The correlation's different. Well, in this case,
they're not even white, because I didn't make
sure they're white. It's very complicated. If you apply the right
techniques, which are techniques of
multi-fractal analysis which is developed for the
purpose of measuring roughness, then all these
eight are the same. Let me come back now to
this matter of metals I mentioned earlier. So [? Don Pesoga, ?]
who was a metallurgist, was getting from the National
Bureau of Standards-- or maybe it was already
a different name-- sort of finger-shaped
piece of metal. In fact, they came
in large numbers. So the National Bureau
took a piece of metal and had an extraordinary careful
treatment, was very expensive piece of metal. They very carefully treat
it at constant temperature or for a long time,
certainly homogeneous. Then they're cut
into five pieces and sent to different
laboratories who ordered them. We [INAUDIBLE] them
bang, bang, bang, bang. We measured the roughness
as the book of metallurgy said it should be done. And that measurement
was, lo and behold, exactly the same
as the measurement advocate in the
books of finance. That is you approximate
this mess by straight line or by plane or whatever and
took at the root-mean-square deviation from that level. And you assume the
root-mean-square deviation is a measure of non-flatness. Well, it is a measure
of non-flatness. But the horrible
fact was that we had five samples which the
National Bureau of Standards guaranteed to be identical. And the measures of roughness
were all over the landscape. So they didn't measure anything. If you measure the
root-mean-square on these things, you get results
completely all over landscape. These things are
totally different. This suggests an extraordinary
level of variability. However, if then measured
for these fractures when you measure the fractal dimension--
not quite as simple as what I wrote to you, but the next
simplest algorithm which makes it measurable-- we found in all cases, lo and
behold, hold your breath, 4/3. Well, not quite 4/3-- 4/3 for lines. And it's normal dimensional
line is one, a curve. So it's a plus 1/3. And for surface,
it was 2 plus 1/3. And then people
redid the experiment on very much longer samples. And they found that,
indeed, the 4/3 [INAUDIBLE] were five orders of magnitude. That is here is a
case of roughness in the most, if you forgive
me, the roughest form, the roughness of
broken piece of metal. By looking at the
wrong measurements, you get values all
over landscape. By looking at the
right measurement, you find that a
certain value can be measured over five or
more orders of magnitude. The heat treatment only changes
the crossovers at the end and makes this range
bigger or smaller. But the precision of that
is absolutely incredible. It's also true for glasses. It's true for a variety
of extraordinary different surfaces. Rough surfaces have this
property of plus 1/3. Now, I view that has
being, first of all, an example of a mixture of
nature reduced to a manageable, concise, and clean conjecture. It is not a
complicated statement about structure of metals. It's a simple statement. That thing is 4/3. And it is, of course, the
greatest clear success, greatest token of
success for a model, which says that this is
the way to measuring it. Now, after having given you
this example, this construction, I want to describe
these diagrams. In physics, it's very
important that once you have a certain model to
change some characteristics, temperature and something
else, into a phase diagram in which one sees how
the properties of a system vary. And so these are three ways of
showing it for the symmetric-- again, coming back the prices
of symmetric generators. And, well, let me show you
this and then come back to the previous thing. This is, on the
left and the right, a test which tests whether
price changes indeed [INAUDIBLE] follow the multi-fractal model. Here, the price in question
is the exchange rate between the dollar
and Deutsche Mark which had the advantage of being
a rather clean finite sample, since on January
1st it will stop, since the Deutsche
Mark will vanish. But it is known for many
periods with excruciating precision every 20 seconds. You can have data in
this context which are data like one never
thought one could find in the context of
economics or finance, the data like in physics
in the best cases. So one does this diagram. And the tau of q is the
conventional name now for the slopes of that thing. I don't want take time for that. On the right, there is this
envelope of these lines. And this envelope is
called f of alpha. And these things are the
fundamental characteristics of the processes. So to come back to
these, if one did the same analysis
on these eight, you find the same results. Again, every analysis
[INAUDIBLE] you find very different results. So here is this matter of a
Manichaean view of the world. When I described to you in
the beginning that rough versus smooth, of
course I have in mind the fact that even
though smoothness is so rare in our experience it
was transformed into geometry in a marvelous fashion. And I was, myself as a
young man, totally immersed in classical geometry, the
most wonderful thing one could imagine. And by this kind of
extraordinary miracle, it gave rise to a description
of nature beginning by apsis being good for Kepler
for describing planet's motion, et cetera, et cetera. The whole of science is
dominated by a mathematics which is that of the smooth. And that mathematics
arose early on. Derivatives, of course, are
part of it very strongly. And most differential
manifolds and so on, you find these words
differentiable all the time. These are the primary objects of
mathematics and, in particular, of science in
gravitation and light. But, now, we have
the situation that in some parts of nature things
don't fit in this mold at all. So by this belated
process, which started with very, very
abstract mathematics, was trying to get
separate from physics. Again, remember these
words of Cantor, which I find totally
extraordinary, that mathematicians can
free themselves from nature. By trying to free
themselves from nature, they introduce these
concepts and related ideas which were called mathematical
monsters and the like. Poincaré was making fun
of Cantor and [INAUDIBLE] by saying that in the past
mathematicians were inventing new shapes to help
us understand nature, but today they invent new
shapes just to annoy me. [LAUGHTER] Well, it was the case. And so the central point
of fractal geometry is that actually Cantor was
just plain dead wrong, that these shapes were not-- the way they're
invented is irrelevant. Whether he knew of decorative
schemes or not is irrelevant. These mathematics was
actually necessary to take the first step for
study of roughness. And so that brings, again,
into very strong relief this question of averaging,
which destroys randomness. The classical pattern was
that of thermodynamics. I don't know that anybody
teaches that any longer or classical thermodynamics
in which randomness [? didn't ?] exist. And then there was
statistical thermodynamics in which energy
was not a constant, but was fluctuating
a little bit-- but so little. Only refined
electronic measurements could pick the thermal noise. And that was discovered
only rather recently, namely about the
time I was born. For everybody, that is the
definition of recently. But the phenomena we deal
here are not like that. Things don't average out. And so what I have developed
increasingly over the years gradually, but focus
now very strongly, is-- which I would like to
give you more or less to finish this talk-- of states of randomness. Now, in mechanics, we all agree
that the laws of mechanics are unique. Well, forget about
details for the phenomena we are looking at. It's all classical
mechanics, Newton and so on. Laws are unique. However, if you look at
an assembly of molecules, it matters very much whether
the temperature and pressure as such, that the whole thing is
a liquid, a gas, or a crystal, or a solid as I'm
used to saying. It matters very greatly. So one does not begin
by studying everything at the initial. When first it's a gas, I
know which tools to use. It's a liquid. I know which tools use
even though liquids are pretty horrible stuff. And solid, I know
which tools to use. Now, in randomness, this
was not at all emphasized that there is a beautiful
axiomatic good for everybody. But then once you go to
practice, especially practice in physics and
engineering and finance, one always look to phenomenon
which averages everything so they're averaged out
to limits very quickly. There was no randomness
after a certain time. But the phenomena which I
study, whether it is in nature-- and that means
turbulence or whatever-- or in culture that is
stock market or everywhere, these phenomena do not give any
evidence of being averaging. I call them widely random. And the very simple
formulas in that context give very, very
complicated structures. I would like to end
up by one thing. I said I was ending, but let
me just add another point. At one point, I was very much
attacked by my friends, mostly my friends. My enemies did not even bother. Because I was describing an
alternative view of nature, which was so different from
the view based on smoothness. Smoothness include,
of course, everything which is ruled by
partial differential equations, Laplace, Fourier,
Poisson, Levy, Stokes. And this thing had
no differential, no derivatives, had strange
behavior, and so on. So do these things belong
to different universes? Well, I don't think you do. I think it's just very unified. I think that partial
differential equations, if pushed beyond the stringent
conditions under which solutions are unique and
smooth and differential and everything, they give rise
to very complicated behaviors. And one example of
it is that if you take a large assembly
of particles interacting by inverse square root
attraction a la Newton, if you start with a Poisson
distribution or just a lattice and put some pinch of
energy to start with, after a very short time, you
see that these particles become extraordinary complicated
and reassemble themselves into a fractal universe. I didn't mention fractal
universe until now, but I will like to say
two words about it. The solution of galaxies in
this universe is fractal. There is no density. All kinds of things occur
which are quite new. Well, that would suggest that
the clustering of galaxies could very well not be due to
specific complicated forces, which we don't know, but the
simple behavior of particles under 1 over r
square attraction. And so that raise the question,
is clustering up there in the sky? Or is it up there in the head? Well, it may well
be in your head. I don't know, but
it's a question which has to be discussed. In fact, one finds that
many of these questions become very, very much revised. Now, if one begins to
include the fractal of the universe-- oh, yes. It's a fact. Early on, which is sort of 20
years ago when I wrote my book, the universe was
expected to be fractal up to 5 megaparsecs, 50
million light years depth. It's nothing by
universal standards. But then it crept
to 20 to 50 to 100. I don't know. It's between 200 and
300 for most people now. And many people believe it's
1,000, which is pretty far. So if it is true that
the universe is fractal up to that level,
then fractality will insert itself
into cosmology in a very strong fashion. Therefore, not only
fractality will be a result of very
simple laws of physics, like 1 over r squared
possibly, but will affect our understanding
about [INAUDIBLE].. Well, I must stop here, because
things got more complicated, become very complicated. And this is not
for this lecture. I've been writing on this topic
for a very long time, as you see. And, also, for those who
are interested in finance, the Journal of
Quantitative Finance has several papers in
the current year on that. I publish right and left. That's my book on finance,
my book on [INAUDIBLE] noise. I would like to end by this
one, which is coming out in a few weeks or a few
days perhaps by Frame and myself on
mathematics education. We find to our utter amazement
that humans like fractals. [LAUGHTER] And I won't mention
anything about art, but this was the result
of a very, very, very poor programmers work. He had some bug, a semicolon
or something dreadful. It gives the result. He start apologizing,
fearing to be fired. I start thanking from
bottom of my heart. Before correcting the bug,
we had his picture taken. It's something around the
Mandelbrot set mistake was made. But it's a whole
topic unto itself that humans seem to be
happy with roughness. You must have the same
experience as I do. Because you say a
mathematician, oh, people say, I find mathematics
utterly dry and dull. Well, I don't, but they do. But this is one
part of mathematics. And that's mathematics of
the flat and the smooth. The mathematics of the rough,
as developed by many people 100 years ago and now it's galloping
development on all sides, never gets this reaction. In fact, they say, if only
I had that in high school, I would have loved mathematics. Well, you don't need to be
told that, that mathematics is lovable. But it's only at
my old age that I dare present my life
and the story of science as being a conflict between two
forces, one force good and bad, the evil. You choose which one's
which, smooth and rough. But I think it has illuminated
my view of how things occurred. And the reason why roughness
came so late, why so difficult, is because it simply
is more difficult. It requires difficult
mathematics which was developed only
very recently, which is developed now for the
purpose of this investigation. It requires just a
different viewpoint. But it is, I think,
definitely a frontier of scientific investigation. Thank you. [APPLAUSE] MODERATOR: Thank you,
Professor Mandelbrot. We have time for
a few questions. We have two microphones
up here, but you've managed to fill in the
density of the auditorium sufficiently high that we
may have some trouble getting to them. So if you want to ask without
the benefit of amplification, please speak out
as well as you can. Do we have any questions? One over here AUDIENCE: [INAUDIBLE]
Mandelbrot, as a high school teacher who is trying to
get all students to be good mathematicians, to
apply themselves and so on, how much of practical
[INAUDIBLE] can I do? MANDELBROT: [INAUDIBLE]. I think much more than
you think and much more than I thought not so long ago. We have now for
several years at Yale a workshop during the summer
for high school teachers. So we get the cream
of Connecticut. So they are the best. But they tell us stories
which are quite incredible. And they ask us about
how early can I bring it. Is junior high right time? Some people say
elementary school. One of our friends--
we have a kind of network of friends
around who have been using these techniques. Some of our friends have tried
with the kindergarten children. They put them blocks. The blocks had O or I or X.
And children played with that. Then you put other
fractal things. And the challenge is to
rebuild the fractal by putting these things in order. The kids become so
absorbed, apparently, that they are lost in. Because that's a real task. It is not a task which you,
the teacher, has imposed. It's a task that nature has
imposed on humanity forever and forever. And so in junior
high, it's very easy. Everybody who's
anybody knows how to promote at 11, which puts
me to shame because I never did learn to program. So it should be early. In a certain sense, the
question of whether this is the right
mathematics to begin with is the wrong question,
because the mathematics with smooth had
been so elaborated, had become very, very
far from experience. And the questions it asks
are so refined that it is not questions which come naturally. They're not questions
about anything [INAUDIBLE] had thought. The Pascal theorem
about the circles-- I was a geometry nut. I loved that. But I was the only
one who loved it. For all the others, it
was artificial thing that somebody names Pascal
just meant to bother them. But the question about
mastering roughness which I have to-- again, the
rule and not the exception. Smoothness is engineering. This table is smooth,
well, not quite so smooth, because it's already
several years there. So it has some roughness,
I'm sure fractal. But with approximation,
it's smooth. It's engineering,
which is cold, which is dry, et cetera, et cetera. So if mathematics can
go back to its origins-- and it is not even something
which is artificial. For example, I mean,
give an example why. I'm discussing that with
a friend the other day. I introduced a notion
which was taken as being the end of madness,
that of negative dimension. Now, you measure things,
so you measure roughness. And, now, you can also
measure vacuity, emptiness. So listen to that. If you take two lines
in [INAUDIBLE] space, their intersection-- well, they [INAUDIBLE]
intersect, of course. But intersection
of lines is less empty than intersection of line
and a point or of two points. That looks like science fiction. Actually, it's a question which
arose when friends of mine were looking at intermittency
of turbulence in the laboratory, because-- I could explain why. It's a long story. But then to explain
it, we must understand that lines are not lines. Lines the little tubes. And lines being little tubes is
not the only invention of me. It's the invention
of Minkowski, who was a very great man in
1900 and who point out that one gets away from all
kinds of horrible paradoxes in mathematics by never
thinking of lines are lines or of surface as surface, but
thinking of lines as being tubes, surfaces as
being [INAUDIBLE] et cetera, et cetera. So one goes back
to these things. Not only one goes back to Mother
Earth to get strength again, but one does in a fashion
which are fruitful. Because without this
element of going back into epsilon neighborhoods,
to call them by a fancy term, one could not make sense
of the negative dimension. And negative dimension is
something you can measure and which is a useful thing
in the study of turbulence. MODERATOR: Do we have
another question? AUDIENCE: Are there any human
behavior in fractal terms? MANDELBROT: Well, certainly,
stock market is human behavior. As a matter of fact, I have
come to use the word culture in a sense, which is perhaps
not smooth to some ears speaking fractal
geometry of nature and then giving an
example like stock market. But take example of internet. It's human behavior. It's not exactly what you
think of human behavior. But a large number of
very brilliant people working at cross purposes
put this thing together. And it works marvelous
most of the time. And every so often,
it's terrible and your messages
don't go through. So people first try to
apply to the internet the techniques which had worked
for telephones with Poisson behavior and so on. It was absolutely off the mark. It is a multi-fractal. And it has to be multi-fractal. Now, after the fact,
it's understood. So here is human behavior. I mean, because design of
this very informal, and noisy, and messy design of
this huge system, which for reasons which
are, after the fact, sort of vaguely understood,
is multi-fractal. But you don't have to
understand it to live with it, because you have
to live with it. And so if you're
making equipment, you better test whether
its properties behaves well in the face of it. But I was trying to start
with human behavior, which is not usually called as such. But art is certainly
human behavior. And so does art have
fractal aspects? Certainly. And let me give you an
example of that, which I still find mind-boggling. My friend Richard Voss
found experimentally that music is 1 over F
noise like these things I was showing you here. He's always a physicist. And he is not a loud person,
so it didn't become very widely known. But then I was
approached independently by two composers, one in
New York and one in Europe. And you must know the names,
Charles Warren and Gyorgy Ligeti, who told me that
looking at my pictures made them understand
nature of their craft. And I asked for
each day, how come? And they say, well, I-- I speak in the voice of
either, because exactly the same explanation. I have been trained in a
very classical conservatory tradition, so I knew very well
about all the instruments, their values, about everything. But I was not told
one basic thing, which is what distinguishes
a piece of music from a collection of noises. And I learned, each of them
said, by trial and error. I brought a composition
to my teacher. He said, oh, too busy. And I brought
another composition. It's not decorated enough. And another proposition, it
just goes up and down too much, doesn't go up and not too much. After a while, I finally
understood what to do. And I've been doing
that very well. And I am sure that you
know Ligeti, and Warren. They are very famous people. And then I look at the picture,
I say, but it's obvious. It's obvious, except
nobody told me that. It's multi-fractal. That is a sonata, 21 minutes,
three movements, allegro, lento, presto, different. Each movement-- loud,
soft, very loud. So the thing must have
structure at all scales. Something will be
changing at every scale. And that's it. If you get that, you
get at least bad music. [LAUGHTER] And bad music is much
better than noise. And everybody think by music. And Voss was telling
a story about he did some pentatonic
music and the Chinese say it's bad Korean music,
Korean bad Japanese music, Japanese bad Thai
music, whatever, but it was music of
some barbarian race. So that essence, it's not
at all fundamental, again, good and bad music
[INAUDIBLE] together. Now, so I can go on and on. Painting-- so in the
classical landscape, which is very artificial form of art
both in oriental and European tradition, so there was big
tree which framed it and then this little man and so on. There are all these
rules of composition. And you look at it-- one big thing. And most of these books of
art did speak of design. And I tell you what
I did discover. Oh, I do my own
covers for my books. It's kind of-- well,
I like to do it. So this book for teaching
has every single cliche of composition, because I
felt it would be effective. But on the other
hand, I was looking at the film of Kandinsky
painting on his paintings. It looked exactly like a program
to do fractals would look. He would look at
this piece of a thing and then boom, a big
line, solid line. You see that in all
kinds of composition. Then he would set small lines. When the film ends,
has a small brush and he's adding little
things all over the place. So he puts big and small
and medium and so on elements of scale
which is exactly the kind of basic
bottom multi-fractality. I could go on
forever and forever. This, I'm speaking
of current behavior. But if you look at
this African village-- so this behavior is not in
the sense of a physiologist, but a sense of the artist. Now, physiologist,
that's another matter. Now, many of my friends
tell me that they have a [INAUDIBLE]
recordings look like these things I showed. I'd be very much
interested in seeing them. But I know very well that if
you describe some surfaces in the body that is
certainly human behavior. Some surfaces are meant
to be as small as possible given their volume and
others as big as possible. The skin should be
smooth and taught, like a young child's skin. And the lung inside must
be as confused as possible. So the two criteria of design
lead to either smooth shapes or to extremely convoluted
shapes, which happen to have fractal features. The human lung, for example, is
branching fractal on 23 levels. 23 levels is a very
healthy bifurcation. It's a very healthy number. It's a [? million. ?] MODERATOR: I think we're going
to need to wrap up right now, but I'll invite all of you who
want to stay to come on down and speak with Professor
Mandelbrot for a few minutes. And I'd like to conclude
by thanking Professor Lorenz for joining us today. [APPLAUSE] And let's give thanks
to Professor Mandelbrot. [APPLAUSE]
