[Paleopsych] Edge: A Theory of Roughness: A Talk with Benoit Mandelbrot
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That's eight for today. You're almost certain to get a full ten tomorrow,
since that's when the Chronicle of Higer Education comes out and when I
check Arts & Letters daily, my main sources besides the New York Times.
A Theory of Roughness: A Talk with Benoit Mandelbrot
http://www.edge.org/3rd_culture/mandelbrot04/mandelbrot04_index.html
A recent, important turn in my life occurred when I realized that
something that I have long been stating in footnotes should be put on
the marquee. I have engaged myself, without realizing it, in
undertaking a theory of roughness. Think of color, pitch, loudness,
heaviness, and hotness. Each is the topic of a branch of physics.
Chemistry is filled with acids, sugars, and alcohols -- all are
concepts derived from sensory perceptions. Roughness is just as
important as all those other raw sensations, but was not studied for
its own sake.
.
A THEORY OF ROUGHNESS
A Talk with Benoit Mandelbrot
Introduction
During the 1980s Benoit Mandelbrot accepted my invitation to give a
talk before The Reality Club. The evening was the toughest ticket in
the 10 year history of live Reality Club events during that decade: it
seemed like every artist in New York had heard about it and wanted to
attend. It was an exciting, magical evening. I've stayed in touch with
Mandelbrot and shared an occasional meal with him every few years,
always interested in what he has to say. Recently, we got together
prior to his 80th birthday.
Mandelbrot is best known as the founder of fractal geometry which
impacts mathematics, diverse sciences, and arts, and is best
appreciated as being the first broad attempt to investigate
quantitatively the ubiquitous notion of roughness.
And he continues to push the envelope with his theory of roughness.
"There is a joke that your hammer will always find nails to hit," he
says. "I find that perfectly acceptable. The hammer I crafted is the
first effective tool for all kinds of roughness and nobody will deny
that there is at least some roughness everywhere."
"My book, The Fractal Geometry of Nature," he says, reproduced
Hokusai's print of the Great Wave, the famous picture with Mt. Fuji in
the background, and also mentioned other unrecognized examples of
fractality in art and engineering. Initially, I viewed them as amusing
but not essential. But I soon changed my mind.
"Innumerable readers made me aware of something strange. They made me
look around and recognize fractals in the works of artists since time
immemorial. I now collect such works. An extraordinary amount of
arrogance is present in any claim of having been the first in
"inventing" something. It's an arrogance that some enjoy, and others
do not. Now I reach beyond arrogance when I proclaim that fractals had
been pictured forever but their true role had remained unrecognized
and waited for me to be uncovered."
[10]
-- JB
BENOIT MANDELBROT is Sterling Professor of Mathematical Sciences at
Yale University and IBM Fellow Emeritus (Physics) at the IBM T.J.
Watson Research Center. His books include The Fractal Geometry of
Nature; Fractals and Scaling in Finance); and (with Richard L. Hudson)
The (mis)Behavior of Markets.
[11]Benoit Mandelbrot's [12]Edge Bio Page
_________________________________________________________________
A THEORY OF ROUGHNESS
(BENOIT MANDELBROT:) There is a saying that every nice piece of work
needs the right person in the right place at the right time. For much
of my life, however, there was no place where the things I wanted to
investigate were of interest to anyone. So I spent much of my life as
an outsider, moving from field to field, and back again, according to
circumstances. Now that I near 80, write my memoirs, and look back, I
realize with wistful pleasure that on many occasions I was 10, 20, 40,
even 50 years "ahead of my time." Until a few years ago, the topics in
my Ph.D. were unfashionable but they are very popular today.
My ambition was not to create a new field, but I would have welcomed a
permanent group of people having interests close to mine and therefore
breaking the disastrous tendency towards increasingly well-defined
fields. Unfortunately, I failed on this essential point, very badly.
Order doesn't come by itself. In my youth I was a student at Caltech
while molecular biology was being created by Max Delbrück, so I saw
what it means to create a new field. But my work did not give rise to
anything like that. One reason is my personality -- I don't seek power
and do not run around. A second is circumstances -- I was in an
industrial laboratory because academia found me unsuitable. Besides,
creating close organized links between activities which otherwise are
very separate might have been beyond any single person's ability.
That issue is important to me now, in terms of legacy. Let me
elaborate. When I turned seventy, a former postdoc organized a festive
meeting in Curaçao. It was superb because of the participation of
mathematician friends, physicist friends, engineering friends,
economist friends and many others. Geographically, Curaçao is out of
the way, hence not everybody could make it, but every field was
represented. Several such meetings had been organized since 1982.
However, my enjoyment of Curaçao was affected by a very strong feeling
that this was going to be the last such common meeting. My efforts
over the years had been successful to the extent, to take an example,
that fractals made many mathematicians learn a lot about physics,
biology, and economics. Unfortunately, most were beginning to feel
they had learned enough to last for the rest of their lives. They
remained mathematicians, had been changed by considering the new
problems I raised, but largely went their own way.
Today, various activities united at Curaçao are again quite separate.
Notable exceptions persist, to which I shall return in a moment.
However, as I was nearing eighty, a Curaçao-like meeting was not
considered at all. Instead, the event is being celebrated by more than
half a dozen specialized meetings in diverse locations. The most novel
and most encouraging one will be limited to very practical
applications of fractals, to issues concerning plastics, concrete, the
internet, and the like.
For many years I had been hearing the comment that fractals make
beautiful pictures, but are pretty useless. I was irritated because
important applications always take some time to be revealed. For
fractals, it turned out that we didn't have to wait very long. In pure
science, fads come and go. To influence basic big-budget industry
takes longer, but hopefully also lasts longer.
To return to and explain how fractals have influenced pure
mathematics, let me say that I am about to spend several weeks at the
Mittag-Leffler Institute at the Swedish Academy of Sciences. Only 25
years ago, I had no reason to set foot there, except to visit the
spectacular library. But, as it turned out, my work has inspired three
apparently distinct programs at this Institute.
~~
The first was held in the 1980s when the Mandelbrot Set was a topic of
a whole year of discussion. It may not be widely appreciated that the
discovery of that set had consisted in empowering the eye again, in
inspecting pictures beyond counting and on their basis stating a
number of observations and conjectures to which I drew the
mathematicians' attention. One of my conjectures was solved in six
months, a second in five years, a third in ten. But the basic
conjecture, despite heroic efforts rewarded by two Fields Medals,
remains a conjecture, now called MLC: the Mandelbrot Set is locally
connected. The notion that these conjectures might have been reached
by pure thought -- with no picture -- is simply inconceivable.
The next Mittag-Leffler year I inspired came six years ago and focused
on my "4/3" conjecture about Brownian motion. Its discovery is
characteristic of my research style and my legacy, hence deserves to
be retold.
Scientists have known Brownian motion for centuries, and the
mathematical model provided by Norbert Wiener is a marvelous pillar at
the very center of probability theory. Early on, scientists had made
pictures both of Brownian motion in nature and of Wiener's model. But
this area developed like many others in mathematics and lost all
contact with the real world.
My attitude has been totally different. I always saw a close kinship
between the needs of "pure" mathematics and a certain hero of Greek
mythology, Antaeus. The son of Earth, he had to touch the ground every
so often in order to reestablish contact with his Mother; otherwise
his strength waned. To strangle him, Hercules simply held him off the
ground. Back to mathematics. Separation from any down-to-earth input
could safely be complete for long periods -- but not forever. In
particular, the mathematical study of Brownian motion deserved a fresh
contact with reality.
Seeking such a contact, I had my programmer draw a very big sample
motion and proceeded to play with it. I was not trying to implement
any preconceived idea, simply actively "fishing" for new things. For a
long time, nothing new came up. Then I conceived an idea that was less
scientific than esthetic. I became bothered by the fact that, when a
Brownian motion has been drawn from time 0 to time 1, its two end
portions and its middle portion follow different rules. That is, the
whole is not homogeneous, exhibits a certain lack of inner symmetry, a
deficit of beauty.
This triggered the philosophical prejudice that when you seek some
unspecified and hidden property, you don't want extraneous complexity
to interfere. In order to achieve homogeneity, I decided to make the
motion end where it had started. The resulting motion biting its own
tail created a distinctive new shape I call Brownian cluster. Next the
same purely aesthetic consideration led to further processing. The
continuing wish to eliminate extraneous complexity made me combine all
the points that cannot be reached from infinity without crossing the
Brownian cluster. Painting them in black sufficed, once again, to
create something quite new, resembling an island. Instantly, it became
apparent that its boundary deserved to be investigated. Just as
instantly, my long previous experience with the coastlines of actual
islands on Earth came handy and made me suspect that the boundary of
Brownian motion has a fractal dimension equal to 4/3. The fractal
dimension is a concept that used to belong to well-hidden mathematical
esoteric. But in the previous decades I had tamed it into becoming an
intrinsic qualitative measure of roughness.
Empirical measurement yielded 1.3336 and on this basis, my 1982 book,
The Fractal Geometry of Nature, conjectured that the value of 4/3 is
exact. Mathematician friends chided me: had I told them before
publishing, they could have quickly provided a fully rigorous proof of
my conjecture. They were wildly overoptimistic, and a proof turned out
to be extraordinarily elusive. A colleague provided a numerical
approximation that fitted 4/3 to about 15 decimal places, but an
actual proof took 18 years and the joining of contributions of three
very different scientists. It was an enormous sensation in the year
2000. Not only the difficult proof created its own very active sub
field of mathematics, but it affected other, far removed, sub fields
by automatically settling many seemingly unrelated conjectures. An
article in Science magazine reported to my great delight a comment
made at a major presentation of the results, that this was the most
exciting thing in probability theory in 20 years. Amazing things
started happening and the Mittag-Leffler Institute organized a full
year to discuss what to do next.
Today, after the fact, the boundary of Brownian motion might be billed
as a "natural" concept. But yesterday this concept had not occurred to
anyone. And even if it had been reached by pure thought, how could
anyone have proceeded to the dimension 4/3? To bring this topic to
life it was necessary for the Antaeus of Mathematics to be compelled
to touch his Mother Earth, if only for one fleeting moment.
Within the mathematical community, the MLC and 4/3 conjectures had a
profound effect -- witnessed recently when the French research
council, CNRS, expressed itself as follows. "Mathematics operates in
two complementary ways. In the 'visual' one the meaning of a theorem
is perceived instantly on a geometric figure. The 'written' one leans
on language, on algebra; it operates in time. Hermann Well wrote that
'the angel of geometry and the devil of algebra share the stage,
illustrating the difficulties of both.'"
I, who took leave from French mathematics at age 20 because of its
rage against images, could not have described it better. Great to be
alive when these words come from that pen. But don't forget that, in
the generations between Hermann Well (1885-1955) and today -- the
generations of my middle years -- the mood had been totally different.
Back to cluster dimension. At IBM, where I was working at the time, my
friends went on from the Brownian to other clusters. They began with
the critical percolation cluster, which is a famous mathematical
structure of great interest in statistical physics. For it, an
intrinsic complication is that the boundary can be defined in two
distinct ways, yielding 4/3, again, and 7/4. Both values were first
obtained numerically but by now have been proven theoretically, not by
isolated arguments serving no other purpose, but in a way that has
been found very useful elsewhere. As this has continued, an enormous
range of geometric shapes, so far discussed physically but not
rigorously, became attractive in pure mathematics, and the proofs were
found to be very difficult and very interesting.
The third meeting that my work inspired at the Mittag-Leffler
Institute of the Swedish Academy, will take place this year. Its
primarily concern will be a topic I have already mentioned, the
mathematics of the Internet.
This may or may not have happened to you, but some non-negligible
proportion of e/mail gets lost. Multiple identical messages are a
pest, but the sender is actually playing it safe for the good reason
that in engineering everything is finite. There is a very complicated
way in which messages get together, separate, and are sorted. Although
computer memory is no longer expensive, there's always a finite size
buffer somewhere. When a big piece of news arrives, everybody sends a
message to everybody else, and the buffer fills. If so, what happens
to the messages? They're gone, just flow into the river.
At first the experts thought they could use an old theory that had
been developed in the 1920s for telephone networks. But as the
Internet expanded, it was found that this model won't work. Next they
tried one of my inventions from the mid-1960s, and it wouldn't work
either. Then they tried multi fractals, a mathematical construction
that I had introduced in the late 1960s and into the 1970s. Multi
fractals are the sort of concept that might have been originated by
mathematicians for the pleasure of doing mathematics, but in fact it
originated in my study of turbulence and I immediately extended it to
finance. To test new internet equipment one examines its performance
under multi fractal variability. This is even a fairly big business,
from what I understand.
~~
How could it be that the same technique applies to the Internet, the
weather and the stock market? Why, without particularly trying, am I
touching so many different aspects of many different things?
A recent, important turn in my life occurred when I realized that
something that I have long been stating in footnotes should be put on
the marquee. I have engaged myself, without realizing it, in
undertaking a theory of roughness. Think of color, pitch, heaviness,
and hotness. Each is the topic of a branch of physics. Chemistry is
filled with acids, sugars, and alcohols; all are concepts derived from
sensory perceptions. Roughness is just as important as all those other
raw sensations, but was not studied for its own sake.
In 1982 a metallurgist approached me, with the impression that fractal
dimension might provide at long last a measure of the roughness of
such things as fractures in metals. Experiments confirmed this hunch,
and we wrote a paper for Nature in 1984. It brought a big following
and actually created a field concerned with the measurement of
roughness. Recently, I have moved the contents of that paper to page 1
of every description of my life's work.
Those descriptions have repeatedly changed, because I was not
particularly precocious, but I'm particularly long-lived and continue
to evolve even today. Above a multitude of specialized considerations,
I see the bulk of my work as having been directed towards a single
overarching goal: to develop a rigorous analysis for roughness. At
long last, this theme has given powerful cohesion to my life. Earlier
on, since my Ph.D. thesis in 1952, the cohesion had been far more
flimsy. It had been based on scaling, that is, on the central role
taken by so-called power-law relations.
For better or worse, none of my acquaintances has or had a similar
story to tell. Everybody I have known has been constantly conscious of
working in a pre-existing field or in one being consciously
established. As a notable example, Max Delbrück was first a physicist,
and then became the founder of molecular biology, a field he always
understood as extending the field of biology. To the contrary, my fate
has been that what I undertook was fully understood only after the
fact, very late in my life.
To appreciate the nature of fractals, recall Galileo's splendid
manifesto that "Philosophy is written in the language of mathematics
and its characters are triangles, circles and other geometric figures,
without which one wanders about in a dark labyrinth." Observe that
circles, ellipses, and parabolas are very smooth shapes and that a
triangle has a small number of points of irregularity. Galileo was
absolutely right to assert that in science those shapes are necessary.
But they have turned out not to be sufficient, "merely" because most
of the world is of infinitely great roughness and complexity. However,
the infinite sea of complexity includes two islands: one of Euclidean
simplicity, and also a second of relative simplicity in which
roughness is present, but is the same at all scales.
The standard example is the cauliflower. One glance shows that it's
made of florets. A single floret, examined after you cut everything
else, looks like a small cauliflower. If you strip that floret of
everything except one "floret of a floret" -- very soon you must take
out your magnifying glass -- it's again a cauliflower. A cauliflower
shows how an object can be made of many parts, each of which is like a
whole, but smaller. Many plants are like that. A cloud is made of
billows upon billows upon billows that look like clouds. As you come
closer to a cloud you don't get something smooth but irregularities at
a smaller scale.
Smooth shapes are very rare in the wild but extremely important in the
ivory tower and the factory, and besides were my love when I was a
young man. Cauliflowers exemplify a second area of great simplicity,
that of shapes which appear more or less the same as you look at them
up close or from far away, as you zoom in and zoom out.
Before my work, those shapes had no use, hence no word was needed to
denote them. My work created such a need and I coined "fractals." I
had studied Latin as a youngster, and was trying to convey the idea of
a broken stone, something irregular and fragmented. Latin is a very
concrete language, and my son's Latin dictionary confirmed that a
stone that was hit and made irregular and broken up, is described in
Latin by the adjective "fractus." This adjective made me coin the word
fractal, which now is in every dictionary and encyclopedia. It denotes
shapes that are the same from close and far away.
~~
Do I claim that everything that is not smooth is fractal? That
fractals suffice to solve every problem of science? Not in the least.
What I'm asserting very strongly is that, when some real thing is
found to be un smooth, the next mathematical model to try is fractal
or multi fractal. A complicated phenomenon need not be fractal, but
finding that a phenomenon is "not even fractal" is bad news, because
so far nobody has invested anywhere near my effort in identifying and
creating new techniques valid beyond fractals. Since roughness is
everywhere, fractals -- although they do not apply to everything --
are present everywhere. And very often the same techniques apply in
areas that, by every other account except geometric structure, are
separate.
To give an example, let me return to the stock market and the weather.
It's almost trite to compare them and speak of storms and hurricanes
on Wall Street. For a while the market is almost flat, and almost
nothing happens. But every so often it hits a little storm, or a
hurricane. These are words which practical people use very freely but
one may have viewed them as idle metaphors. It turns out, however,
that the techniques I developed for studying turbulence -- like
weather -- also apply to the stock market. Qualitative properties like
the overall behavior of prices, and many quantitative properties as
well, can be obtained by using fractals or multi fractals at an
extraordinarily small cost in assumptions.
This does not mean that the weather and the financial markets have
identical causes -- absolutely not. When the weather changes and
hurricanes hit, nobody believes that the laws of physics have changed.
Similarly, I don't believe that when the stock market goes into
terrible gyrations its rules have changed. It's the same stock market
with the same mechanisms and the same people.
A good side effect of the idea of roughness is that it dissipates the
surprise, the irritation, and the unease about the possibility of
applying fractal geometry so widely.
The fact that it is not going to lack problems anytime soon is
comforting. By way of background, a branch of physics that I was
working in for many years has lately become much less active. Many
problems have been solved and others are so difficult that nobody
knows what to do about them. This means that I do much less physics
today than 15 years ago. By contrast, fractal tools have plenty to do.
There is a joke that your hammer will always find nails to hit. I find
that perfectly acceptable. The hammer I crafted is the first effective
tool for all kinds of roughness and nobody will deny that there is at
last some roughness everywhere.
I did not and don't plan any general theory of roughness, because I
prefer to work from the bottom up and not from top to bottom. But the
problems are there. Again, I didn't try very hard to create a field.
But now, long after the fact, I enjoy this enormous unity and
emphasize it in every recent publication.
The goal to push the envelope further has brought another amazing
development, which could have been described as something recent, but
isn't. My book, The Fractal Geometry of Nature, reproduced Hokusai's
print of the Great Wave, the famous picture with Mt. Fuji in the
background, and also mentioned other unrecognized examples of
fractality in art and engineering. Initially, I viewed them as amusing
but not essential. But I changed my mind as innumerable readers made
me aware of something strange. They made me look around and recognize
fractals in the works of artists since time immemorial. I now collect
such works. An extraordinary amount of arrogance is present in any
claim of having been the first in "inventing" something. It's an
arrogance that some enjoy, and others do not. Now I reach beyond
arrogance when I proclaim that fractals had been pictured forever but
their true role remained unrecognized and waited for me to be
uncovered.
_________________________________________________________________
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References
10. http://www.edge.org/3rd_culture/bios/brockman.html
11. http://www.edge.org/3rd_culture/bios/mandelbrot.html
12. http://www.edge.org/3rd_culture/bios/mandelbrot.html
13. http://www.edge.org/3rd_culture/mandelbrot04/images/curious500.jpg
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