Fractals and the art of roughness Benoit Mandelbrot
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thank you very much please excuse me for
sitting I’m very old
well the topic I’m going to discuss is
one which is in a certain sense very
peculiar because it’s very old roughness
is part of human life forever and
forever and ancient authors have written
about it it was very much on
controllable and in a certain sense it
seemed to be the extreme of complexity
just a mess a mess a mess the many
different kinds of a mess now
in fact by complete fluke I got involved
many years ago in the study of this form
of complexity and to my utter amazement
I found traces very strong cases i’ma
say of order in that roughness and so
today I would like to present to you a
few of examples of what this represents
I prefer to the word roughness to the
word irregularity because irregularity -
somebody had had Latin in my long past
use means the country of regularity but
it is not so regularity is a country of
roughness because the basic aspect of
the world is very rough so let me show
you a few objects some of them
artificial other of them very real in a
certain sense and now this video is a
cauliflower now wider short cauliflower
a very ordinary an ancient vegetable
because all the ancient as maybe it’s
very complicated and it’s very simple
both the same time if you try to wait of
course is very easy to wait and I eat it
than the way it matters but suppose
suppose you try to measure its surface
well it’s very interesting if you’d cut
with short knife one of the florets
electron flower and look additi
separately you think
whole cauliflower was smaller and then
you cut again again again again again
again again again and still gets more
cauliflower so the experience of
humanity has always been that there are
some shapes which have this peculiar
property that each part is like the
whole but smaller now what did you want
to do with that very very little so what
I did actually stoom is to study the
this this problem and found something
quite quite surprising that one can
measure roughness by number number two
point three one point two and sometimes
much more when one day a friend of mine
to bug me he brought the pictures it
what is the roughness of this this curve
I said well just short of one point five
was 1.48 now it didn’t in any time I’ve
been looking at these things for so long
so these numbers are the numbers usually
denoted the roughness of the these
surfaces hasten to say that each surface
is completely artificial done on a
computer and the only input is a number
and that number is roughness and so on
the left
I took it the roughness copied from many
landscapes to the right
I took a higher roughness so the eye
after a while can distinguish these two
very well humanity had to learn about
measuring roughness this is very rough
and this is sort of smooth and is
perfectly smooth very few pieces are
very smooth so then he’ll try to to ask
questions how was surface a cauliflower
well you you measure a measure measure
each term look closer it gets bigger
down to very very small distances what’s
the length of coastline of the legs the
closer to measure the longer it is the
concept of length across line which
seems to be so natural because it’s
given in many places is in fact from
fallacy there’s no such thing you must
do it differently what good is that to
know these things well surprisingly
enough is good in many ways to begin
with especially landscapes which I
invented sort of are used in in cinema
all the time we see motifs in distance
they may be mountains but they may be
just formula just blanked on now it’s
very easy to do it was used to be very
time-consuming but now it’s nothing now
look at that
that’s real longer now along with
something very strange if you take this
thing you know very well it weighs very
little the volume of a long is very
small but what about the area of the
lung anatomist were arguing very much
about that some said that normal normal
males long has an area inside of a
basketball and the other said no five
basketballs enormous disagreements why
so because in fact the area of the lung
is something very ill-defined the the
bronchi branch from a tree branch and
the branch not because in the stop
branching not because of of any matter
of principle but because of physical
concept consideration democracy which is
in the longer so what happens is that
way we have much bigger long but if it’s
branches the branches down to distance
is about the same for well for man and
for a little rodent so now what good is
it to have that well surprisingly enough
amazingly enough the anatomist had a
very poor idea of the structure of the
long until very recently and I think
that my mathematics surprisingly enough
has been a great help to them to the
surgeons studying along illnesses and
also kidney illnesses all these
branching systems which were for which
Tourneau geometry so I found myself in
other words constructing a geometry a
geometry of things which had no geometry
and surprising as the
it is that very often the rules of this
geometry are extremely short you have
formulas that long any client it several
times sometimes repeatedly again again
again the same a petition and at the end
you get things like that this cloud is
completely 100% artificial well ninety
nine point nine and the only part which
is natural is a number the roughness of
the cloud which is taken from nature
something so complicated cloud so
unstable so varying should have a simple
rule behind it now the simple rule
doesn’t is not exponential cloud and the
sea of clouds had to take account of it
I don’t know how how it much had
advanced these pictures are the old I
was very much involved in it but then
turn my attention to other phenomena now
here is another thing which is rather
interesting one of the shattering events
in the history of mathematics which is
not appreciated by many people occurred
about ten thirty years ago around forty
five years ago mathematicians began to
create shapes that didn’t exist
mathematicians brought into into
self-praise an extent which was
absolutely amazing that man can invent
things that nature did not know in
particular it could invent things like a
curve which fills the plane curve the
curve a plane the plane and to won’t mix
well they do mix a man named piano did
define such curves and it became an
object of extraordinary interest it’s
very important but mostly interest
because a kind of break a separation
between the mathematics coming from
reality on the one hand a numismatist
coming from pure minds mind well I was
very sorry to point out that the pure my
man’s mind has in fact seen at long last
what had been seen for a long time and
so here I introduce something the set of
rivers of a plane
filling curve and well it’s a story unto
itself so it was in 1825 825 an
extraordinary period in which
mathematics prepares itself to break out
from the world and the objects which
were used as examples when I was a child
and an A student as examples of the
break between mathematics and visible
reality those objects I turned them
completely around I use them for
describing some of the aspects of the
complexity of nature when a man named
hausdorff in 1919 introduced a number
which was a just mathematical joke and I
found that this number was a good
measurement of roughness when I first
told my friends in mathematics I said oh
don’t don’t be silly it’s just something
well actually I was city gate painter
hawks I knew it very well
the things on the ground are algin he
did not know the mathematics it didn’t
exist and he was Japanese we didn’t have
no contact with the West but painting
for a long time had the flat side that
we speak of that for a long time the
Eiffel Tower has a fractal aspect and I
read the book that mr. Eiffel wrote
about his tower and indeed it was
astonishing how much he understood this
is a mess mess mess
Brian loop one day I decided halfway
through my career I was helped by so
many things in my work I decided to test
myself could I just look at something
which everybody had been looking at for
a long time and find something
geometrically new well as I looked at at
this at this same Cobra emotion just
goes around I made it I played with it
for a while that may return to the
origin then I was telling my assistant I
don’t see anything can you paint it so
he painted it which means that he put
inside everything so when this thing
came artist stop stop stop I see it in
Island and then amazing so Brian motion
which happens to have roughly so number
who goes around I measured it 1.33 again
again again long measurements big bowel
motions 1.33 mathematical problem how to
prove it
it took my friends 20 years three of
them were having incomplete proofs they
brought together and together had the
proof so they got a big metal
mathematics one of the three metals that
people received for proving things we
should have seen without being able to
prove them now everybody had asked at
one point another how did it all start
what brought you in that strange
business what brought you to be at the
same time a mechanical engineer
geographer a mathematician and so on a
physicist well actually I started
oddly enough studying stock market
prices and so here I had this table to
this theory and I wrote books about it
financial crises increments to the left
you see data over long periods with the
right on that on top you see a theory
which is very very fashionable it was
very easy you can write many books very
fast about it so the top traveling’s of
books on that now compare that with the
real price increments and when a real
price increments well these other lines
includes some real price increments and
some forgeries chart data so the idea
there was that one must be able to undo
to how to say a model price variation
and it went see me well fifty years ago
for 50 years people were sort of pulling
this because they would do it much much
much easier but I tell you at this point
people listen to me
this these two curves are averages
thunder and poor the blue one and the
red one is standard pour from which the
five biggest discontinuities are are
taken out now the spot annuities are a
nuisance so in all of many studies of
prices one push them aside
well Axelrod and you have little
nonsense which is left the Axelrod on
this picture as are five Axelrod are as
important and everything else in other
words it is not Axelrod that we should
put aside that is the meet
the problem if you master these you
master price and if you don’t master
this you can master the noise as well as
you can but it’s not important
well here are the curves for it now I
get the final thing which is the set of
which which my name attached them in
anyway it’s a story of my life my
adolescence was spent the German
occupation of France and since I thought
that I might may vanish within a day or
the week I had very big dreams and after
the war I saw an uncle again my own
clothes very prominent a magician he
told me look there’s a problem which I
could not solve twenty-five years ago
and which nobody had to solve this is
construction of man named Julia and
number two if you could if you could
find something new anything he will get
your career mail
very simple so I looked and like the
thousands of people are try before I
found nuttin but then the computer came
and I decided to apply computer not to
new problems mathematics like this legal
because that were new problem to old
problems and that went from what’s
called real numbers which two points
online to imagine complex numbers which
are points in the plane which is what
one should do there and this shape came
out this shape is of an extraordinary
complication the equation is sittin
there Z goes into the square plus C it’s
so simple so drive is so unint
sting now you turn the crank one twice
twice Marvel’s come out I mean this
comes out III don’t want to explain
these things this comes out this comes
out shapes which are of such
complication such harmony and such
beauty this comes out repeatedly again
again again and that was one of my major
discoveries to find that these islands
were the same as a whole big thing more
or less and then you get these extra
neighbor oak decoration all over the
place all that from this little formula
which have has whatever five symbols in
it and then this one the color was added
for two reasons first of all because
these shapes are so complicated that one
couldn’t make any sense of the numbers
and if you plot them you must choose
some system and so my principle has been
to always present the shapes with
different colorings because some
colorings emphasize that another has
data on that it’s so complicated
in 1990 I was in famous you came to see
a prize from the University and a few
days later a pilot was flying over the
landscape and found this thing so where
did it come from obviously from the
extra-terrestrial well so the newspaper
in Cambridge published an article about
that discovery and received the next day
5,000 letters from people saying but
that’s simply Mandelbrot set is very big
well let me finish this shape here just
came out of an exercise in pure
mathematics bottomless wonders spring
from simple rules which are repeated
without end thank you very much
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