[Music] Beno Mandel BR Sterling professor emeritus at y University IBM fellow Emeritus at IDM Research Center well regular geometry the geometry of uid is concerned with shapes which are smooth except perhaps for corners and lines special lines which are singularities but some shapes in nature are so complicated that they have complicate they are equally complicated at the big scale and as you come closer and closer and closer they don't become any less complicated closer and closer or you go farther and farther they remain equally complicated there is never a plane never a straight line never
anything smooth and ordinary the idea is very very vague as I expressed it it's uh it's expression of reality fractal geometry is a new subject and each definition I tried to give for it has turned out to be in appropriate so I'm now being KY and saying they are very complex shapes which look the same from close by and from far away well if you look at a shape like straight line what is remarkable is that if you look at a straight line from close by from far away it is the same it's the straight
line that is a straight line has a property of self similarity each piece of state line is the same as the whole line reduced to a Bigg or small extent the plane again is have the same property for a long time it was widely believed that the only shapes having this extraordinary property are straight line the whole plane whole plan the whole space that in a certain sense self similarity with a dull subject because it reduced to very familiar shapes but that is not the case there are many shapes which are self similar again the
same scene from close by and far away and which are far from being straight or plain or solids and those shapes if which I studied and collected and put together and applied in many many domains I called fractals well historically um a mountain could not be represented except for a few mountains which are almost like cones mountains are very complicated they if you look closer and closer you find greater and greater details if you look away until um you find that bigger details become become visible ible and in a certain sense the same structure appears
at at all scales if you look at Coast lines if you look at them for long for from far away from an airplane um well they you don't see details you see a certain complication as you come closer the complication becomes more local but again continues and you come closer and closer and closer and closer the coastline becomes longer and longer and longer because it has more detail entering in however this detail amazingly enough enters in a certain regular fashion therefore one can study a coastline a geometric object because a geometry of for that existed
for a long time and then I put together and applied it to many the mains the Mandel BR set in a certain sense is a implementation of a dream I had and an uncle of mine had seen I was about 20 I was a student of mathematics but not happy with the mathematics that I was taught in France therefore looking for other topics and my an uncle of mine who was a very well-known pure mathematician wanted me to study a certain Theory which was then many years old 30 years old or something but had in
a way stopped developing when he was young he had tried to get this Theory out of a rout he didn't succeed nobody succeeded so there was a case of two men Julia who a teacher of mine and Fatu who had died had had a very good idea in 1910s and then nothing was happening my uncle was telling me if you look at that if you find something new it be a wonderful thing because I couldn't nobody could I looked at it and found it too difficult I just could see nothing I could do then over
the years I put that a bit in the back of my mind until when day I read an obituary it's an interesting story that I was motivated by obituary obituary of great man namare and in that obituary this question was raised again and um at that time I had a computer I had become quite an expert in using computer for mathematics for physics in many Sciences so I decided that perhaps the time has come to please my uncle 35 years later or something pleasee my uncle and do what my uncle had been pushing me to
do so strongly but I approached this topic in a very different fashion of my uncle my uncle was trying to think of something new idea a new problem a new way of developing the the theory of fat and Julia I did something else I went to the computer and tried to experiment I introduced a very high level of experiment in very pure mathem matics um I was at IBM I had run of computers which then were called very big and powerful but in fact were less powerful than a handheld machine today but I had them
and I could make these experiments the conditions were very very difficult but I knew how to look at pictures in fact the reason why I did not go into pure mathematics earlier was that I was dominated by visual I tried to combine the visual uh Beauty and the mathematics so I look at these pictures for a long time in a very unsystematic fashion just to become acquainted in the kind of physical in fashion with those extraordinary difficult and complicated shapes and tools were extraordinarily difficult they were very computer Graphics did not exist practically but we
had a machine which was do which made things doable and I started finding extraordinary complication extraordinary structure extraordinary beauty of of both theoretical kind mathematical and of visual kind and collected observations of my trip in this unknown territory when I presented that work to to my colleagues it was an explosion of Interest everybody in mathematics had given up for 100 years for 200 years the idea idea that you could from pictures from looking at pictures find new new ideas that was the case long ago in the Middle Ages in the Renaissance in in later periods
but then mathematicians have become very abstract pictures were completely eliminated from mathematics in particular when I was young this happened in a very strong fashion so mathematicians did not even conceive of possibility of pictures being helpful to the contrary I went into a a an orgy of looking at pictures by the hundred machines became a little bit better we had friends who improv them who wrote special software to help me which was wonderful that was the good thing about being at IBM and um and I had this collection of of of observations uh which I
gave to my friends in mathematics for their pleasure and for the simulation ER the extraord fact is that the first idea I had which motivated me in that work is a conjecture a mathematical idea which may or may not be true and that idea is still unproven that is the foundation the what started me and what everybody felt would be easy to prove has so far defeated the greatest efforts by experts be proven in a certain sense it's very very strange because because the object itself is understandable even to a child if the object can
be drawn by a child with new computers with new graphic devices and still the basic idea has not impr proven but the development of of it has been extraordinary then it of course slowed down a bit and now again is going up new people are coming in and they prove extraordinary results which nobody was hoping to Pro prove and I am U astonished and of course very pleased by this development the conjecture itself consists in two definitions mandb set to Alternative definitions which are too technical to be described Blackboard but which are both very simple
and which are assumed naively to be equivalent why they assume so because on the pictures I could not see anything difference the obtaining pictures in one way or another way I couldn't tell them apart therefore I assumed they're identical and I went on studying this pie H I found again many interesting observations uh of which most were very confirmed by many other very very skilled mathematicians but the idea that these two these two conditions definitions identical is still open so that two two definition of man set the usual usual one and another one and they
may theoretically be different people are getting close but have not proven it completely well first of all uh one explanation of that is that the feeling for fractality is not new it is one uh very surprising and extraordinary Discovery I made gradually very slowly by looking again at paintings of the past many painters had a clear idea of what fractal are um take a French classic painter named Busan now he he painted beautiful landscapes completely artificial ones imaginary Landscapes and how did he choose them well he had a balance of trees of loans of houses
in the distance he had a balance of small object big objects big trees in front and this balance of objects at at every scale what was gives to P this special feeling take Hawai a famous Japanese painter of around 1800 he did not have any mathematical training he left no followers because his wave of painting or drawing was not uh was too special to him but it is quite clear by looking at how oxy the ey which had been trained on on fractals that hawy understood fractal structures and again at this balance of big small
and intermediate details and as you come close to this um to these marvelous drawings you find that um you understood perfectly fality but he never expressed it nobody ever expressed it and then the next stage of of Japanese image image um experts did some other things so Humanity has known for a long time what fractals are it's a very strange situation in which an idea which had been which each time I look at all documents has deeper and deeper Roots never how to say gelled never got together until um I started playing with the computer
and playing with with topics which nobody was touching because they were just desperate and hopeless well uh the computers computers uh have been sort of spoken about since early 19th century even before but until electronic computers came which is in reality during World War II or shortly afterwards they could not be used for any purpose in science they were just too slow to to to Limited in their capacity um my chance was that I was myself a very visual person again a mathematician who had not started a very unconventional career because my interest was both
in mathematics and in in the ey and I was at IBM a very primitive computer um picture making machines became available um we had to program everything it was it was a heroic and my friends at IBM who helped me deserve a great thank you ER with these tools I could begin to do things which before have been impossible and I could begin to implement an idea of how a mountain looks like to reduce a mountain which is something monstrously complicated to a very simple idea and how do you do it well you have you
make a conjecture hypothesis about shape of mountains and you don't think about the mathematics of it you just make a picture of it if the picture is believed by everybody to be a mountain then there's something true about it or a cloud it was astonishing when um at one point I got the idea of how to make artificial clouds and um with a collaborator we had the pictures made which were theoretically completely artificial pictures based upon one very simple idea and these pictures everybody views that being clouds people don't believe that they are photographs so
we have certainly found something about true about nature and on the other side the completely artificial shapes the shapes that don't exist in nature which for example the Mand set which was completely came out of the blue out a very simple formula which is about 1 inch long and which gives this endless endless stream of questions and results there what happened is that to everybody's surprise there's a very strong inner resemblance between those shapes and the shapes of nature which have been studying and again I spent half my life roughly speaking doing study of nature
in many of its aspects and half of my life studying completely artificial shapes and the two are extraordinary close in one way both are fractal the the theor of chaos and the fractal are separate but have very strong intersection that is one part of chos theory is geometric expressed by fractal shapes another part of chaos theory is not expressed by fractal shapes another part of fractals does not belong to to Chaos Theory so that are two theories which overlap very strongly and do not coincide one of them chaos theory is based upon behavior of systems
defined by equations equations of motion for example and classical mathematics um and around 1900 p and Adar two great mathematicians over time have realized that sometimes the solutions of very simple looking equations can be extremely complicated but 1900 it was too early to develop that idea it was very well expressed and very much discussed but did not how to say could not could not grow um much later of course with computers this idea came to life again and became very important part of science so both chaos theory and fral have had how to see contacts
in the past when they were both impossible to develop and in a certain sense not ready to be developed um again that they intersect very strongly but they're very distinct well a very strong distinction would be made between chaos and and fractals um for example the rules which generate most of the natural fractals models of mountains of clouds and many other phenomena are involve chains and therefore the they are not at all chaotic in the ordinary sense of the word in ordinary current modern C of the word the chaotic in Old sense of the word
which doesn't have any specific meaning um but the I don't like to discuss these questions of um terms the term chaos came to den note something which was which was very confused it helped it gel but um is it's the use of a Biblical name in a certain sense forces us to to add implications which are not important for in mathematics um that's why I I when I time came to give a name to my work I chose the word word fractal which was new H before that there was no need of a word at
all because again there were only a few undeveloped ideas in very many great minds but when a word became necessary I prefer not to use an old word but to create a new one well it was a it was a very very interesting story at one point A friend of mine a older person told me that saw a paper of mine on new topic and said look benoa I tell you you must stop writing all these papers in that field that field that field that field nobody knows where where you are what you are doing
you just sit down and write a book a short book a clear book A book explaining what I've done so I sat down and wrote a book now the book had no title why because the object the topics I've been studying had not been the object of any Theory whatsoever and there are many words which mean nothing but many fields which have no name because they don't exist so the publisher didn't like this this very ponderous title he said look and a friend of mine another friend told me look you create new field you are
entitled to give it a name so I had had Latin in high school and as as it turned out one of my sons was taking Latin in the United States so there was Latin dictionary in our house which is was an exception I went there and tried to look for a word which fitted what I had been working on and um while I was playing with what fraction uh and uh looked under dictionary for where where the word fraction came from it came from a Latin word which meant how to say disconnected rough and rough
and disconnected it was a very very G the idea of roughness originally in Latin so I started playing with fractus which again made that and coin word fractal first of all I put it in this book object fractal in French as it turned out and then the English translation of the book and then the word took off uh first of all people applied it in ways which I didn't find sensible but had nothing I could say about it so then dictionary started defining it each a little bit differently in a certain sense the word became
alive and independent of me I could scream and say I don't like it but it made no difference and I had once the Curiosity of looking on the web um in different countries um having different language what what is under fractal and found that in one country I would not mention it's a word which has been become applied to some nightclubs a fral nightclub is a kind of nightclub I don't know which because because I haven't been there but and I don't know the language but I guessed from from what I could guess what it
was it's a word which has its own life um it I I gave it the definition but then definition came became too narrow because some objects I want to go fractal did not fit the old definition so some people asked me to whe I still believe in definition of whatever 40 years ago I don't but uh I have no control it's something which which which works by itself self um the fact that very many of adults I know never heard of it but the children have is what gives me particular pleasure because student high school
students even the bright ones are very resistant to to how to impose terms and the combination of pictures and of deep theory that you can look at the picture and find that something some idea about this picture is sensible and then be told that very great scientists either cannot prove it or have taken 40 years to prove it or have had to be several of them together to prove it because was so difficult and it could be seen by child understood by a child that aspect is one which very many people find particularly attractive the
field in mathematics in science definitions are simple but Bare Bones until you get to problem which you understand it takes hundreds and hundreds of pages and number years and years of learning in this case you have this formula you clck it in the computer and from a simple formula in a very short time amazingly beautiful things come out which sometimes people can prove them instantly and sometimes great scientist take forever to prove or don't even succeed in proving well uh what I discovered uh quite a while ago in fact that was my my first major
piece of work is that a model of price variation which which everybody was adopting was very far from being applicable it's a very curious story in 1900 a Frenchman named bashier um who was a student of mathematics wrote a he's on the Ser speculation it was not at all an acceptable Topic in pure mathematics and he had a very miserable life but his his piece of his CES was extraordinary extraordinary in a very strange way it applied very well to a theory of bral motion which is in physics so bashier was was a pioneer of
a very marvelous Essen theory in physics but to economics it didn't apply at all it was very ingenious but B had no data in fact no data were available at that time in 1900 so he imagined an artificial Market in which certain rules will apply unfortunately the theory which was developed by Economist um when computers came up was B Theory it does not account for any of the major effects in economics for example it assumes prices are continuous whereas everybody knows that prices are not continuous so people say well all right there are discontinuities but
there are different kind of Economics that we are going not be concerned with disc continuities because too complicated and only with the part which looks more or less continuous but it turns out that discontinuities are as important or more important than the rest the B assume that each price change is independent of the preceding price change it's a very beautiful assumption but it's completely incorrect because we know very very well especially today that for a long time prices may may vary moderately and then suddenly they begin to vary a great deal so either you say
that the theory changes or you say that the theory which exists is not appropriate well I found that theory was defective on both both grounds that was in 1961 62 I forgot the exact dates when the development B became very very rapid um since since nobody wanted to listen to me I did other things many other things but I was waiting because it was quite clear that my time would have to come and unfortunately it has come that is the fluctuation of the economy stock market and commodity markets today are about as they were in
historical times there was no change which made the stock market different today and it was long ago and the lessons which have drawn from earlier periods do represent today's events very accurately but but the situation is much more complicated than bashier had assumed bashier again was a genius bashier had an excellent idea which happened to be very useful in physics but in economics he just lacked data he did not have an awareness of discontinuity which is essential in this context not have an awareness of dependence which also is essential in this context so his theory
is very very far from What You observe in in reality well um my life has been extremely complicated not by my choice at the beginning at all but later on I was had become used to complication and went on um accepting things that other people would have found very difficult to to accept um I was born in Poland moved to France as a child short before World War II during World War II I was lucky to live in the French equivalent of Appalachia a region which is sort of not very high mountains but very very
poor and Appalachia with C even so poorer than Appalachia of the United States and for many years I was um in high school uh where things were very easy it was a small High School way up in the hills and had mostly private intellectual life I read many books many books very good libraries I read many books and I had dreams of all kind dreams which were in a certain sense how to say easy to make because the future the near future was always extremely threatening it was very dangerous period but since I had nothing
to lose I was dreaming of what I could do then uh the war ended I had had a very very little um playing in taking exam which determines a scientist life in France the two schools both very small one tiny and one small which in a certain sense were the the place that every student wanted to go I had only a few months of finding out how the exam proceeded but I took the exam and um well perhaps because of inherited gifts I did very well in fact I nearly missed barely barely missed being um
number one one in France in both schools and in particular I did very well in mathematical problems the physics I could not guess other things I could not guess but then I had this big Choice should I go into mathematics in a small very school or should I go in a bigger school which in a certain sense would leave me time to decide what I wanted to do first I entered this small school uh when I was uh as a matter of fact number one um of the students who entered then but immediately I left
because that school again was going to teach me something which I did not fully believe namely mathematics separate from everything else it was excellent mathematics French mathematics was very high level but in everything else it was not even present and I didn't want to become a Pure mathematician as as a matter of fact my uncle was one so I knew what a p matician was and I did not want to be a pan I want to do something different not less not more but different namely combine pure mathematics at which I was very good with
the real world of which I was very very curious and so I did not go to E po technique to e normal I went to e po technique it was very um rough decision and U ER the the year when I took this decision remembers my memory very very strongly then for several years I just um was lost a bit I was looking for a good place I spent my time very nicely in many ways um but not fully satisfactorily then I became professor in France but realized that I was not fit I was not
it was a job I should spend my life in fortunately IBM was building a research center I went there for the summer sing for a summer only and during the summer decided to stay it was a very big gamble I lost my job in France I received had a job which was extremely uncertain how long would IBM be interested in research but um but the gamble was taken and very shortly afterwards I had this extraordinary Fortune of stopping at Harvard to give a lecture and learning about the price variation in just the right way at
a time where nobody was looking was realizing that either one needed or one could make a the of price variation other than the year of 1900 which bashier had proposed which was very very far from being a representative of the actual things so I went to IBM and um I was fortunate in being allowed um and being Su successful at to go from field to field which a way was what I've been hoping for I I didn't feel comfortable professor as a student mathematics pure mathematics or as a professor pure mathematics I wanted to do
a little bit of everything and explore the world and IBM let me do so uh I touched far more topics than anybody would have found reasonable I was often told sit settle down stay in one field don't go all the time to another field but I I was just compelled to move from one F to another and the fractal geometry was not an idea which I had early on it was something which developed progressively I didn't choose to go in a topic because of Any how to say compelling reason but because the problems there seem
to be somehow similar ilar the ones I knew how to handle I had experience of this kind of problem and gradually realized that I was truly putting together a new Theory a theory of roughness what is roughness everybody knows what's roughness when was roughness discovered well in the prehistory everything is rough except the the circles how many circles are in nature very very few the straight lines very few shapes are very very smooth but geometry had led them aside because they're too complicated and physics had L aside because they were too complicated one couldn't even
measure roughness so by by luck and by reward for persistence I did find found the Sea of roughness which certainly I didn't expect and expecting to find one would have been pure Madness so the one of the high points of my life was when I really suddenly realized realized that this dream had create had my my late adolescence of combining pure mathematics very pure mathematics with very hard things which have been long a nuisance to scientists and to Engineers that this combination was possible and that I put together this new geometry of nature the fral
geometry of nature well I happen to know this this song it was sent to me I was very impressed by by it and by its popularity in a certain sense it is not which one but the combination when people ask me what's my field I say on one hand I fractalist perhaps the only one the only full-time one on the other hand I had been a professor of mathematics at Harvard at Yale at Yale for a long time um I have been so but I'm not a mathematician only I've been a professor of physics of
Economics a long list each element of this list is normal the combination of these elements is very rare at best and so in a certain sense it is not the fact that I was a professor of mathematics at these great universities or a professor of physics at other great universities or that I received among honor doctorates one in medicine believe it or not and one in in civil engineering um it is the coexistence of these various aspects that in one lifetime it is possible if one takes the kind of risks which I took which are
colossal but taking risks I was rewarded by being able to contribute in a very substantial fashion to a variety of fields to I was able to reawaken and solve some very old problems problems which are so old that in a certain sense they were no longer being pursued nobody I didn't know anybody who was trying to Define roughness of metal fractures it was a hopeless subject but I did it and the whole field which had been created by that in a certain sense the the beauty of what what I happened by extraordinary chance to put
together is that nobody would have believed that this was possible and certainly I didn't expect that to be possible I just moved from step to step to step on late they realize that all these things held together and very late did I see that um that in each field they very old problems could be if not solved at least Advanced or reawakened and therefore gradually very much improved in their understanding
