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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