Benoit B. Mandelbrot, MIT 2001 - Fractals in Science, Engineering and Finance (Roughness and Beauty)

[Music] [Music] indeed a pleasure to introduce to you this afternoon dr. Benoit Mandelbrot who presently a sterling professor of mathematical sciences at Yale University and research fellow emeritus at the IBM Thomas J Watson Research Center personnel vault is truly a man for many nations he was born in Warsaw Poland and in the second decade moved to France Where he subsequently received his doctorate for the Faculty of Sciences at the University of Paris and he spent most of the latter half of his life in this country many people however associated more closely with another country Great

Britain because he asked to the title of one of its 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 short of the stick along the coast and indeed if it were not for the incessant waves of tides so arbitrarily stick short stick could be used to link that go to infinity his continuing pursuit of the phenomenon of self-similarity which he has recognized artistic coastlines but in a wide variety

of natural systems ultimately led him to Coin a word is now familiar to nearly everyone fractals and today his lecture title will be fractals and science engineering and finance roughness and beauty this list of honorary degrees and medals and other honours was far too long for me to have memorized I could read it too but this would cut deeply into the speaking time that is rightfully his so if you need further evidence of the stature that he's attained simply look at her own at the Crowd that had filled every seat ten minutes ago without further

ado then that being to do is professor mantle boss it is indeed a delight to be here among you and in particular delight to be introduced by by Ed Lorenz we met actually around the 64 when a mutual friend Eric Mollick Christiansen told me well there is another man in doghouse not quite the euros but a bit further and so you are you two lost Souls may find some interest in common well Eric was very very very clear sighted because indeed the work of of Priscilla reigns and mine have very many points in common even

though they never quite identically the same the same field now the title we see on the screen here is one which the certain sense summarizes my 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 ask themselves What was the leading sort which permeated all my changes of interest from one field to another and frankly my answers were rather cumbersome it's all lately that I realized that everything was very simple all my life I've been working towards what maple has

become a theory of roughness now let me describe this fennel for this phenomena of roughness if we think of which shapes in our ordinary evidence are smooth well the very few plain perhaps for a child Of amitav man a child of one spot once upon a time in a quite a piece of water without wind a circles well full moon the pupil and the iris of the eye a few more shapes like that straight lines very few on the other side there are 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 on sensation 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 to a science more precisely each went through three stages for example in acoustics the idea of sound was known to from time immemorial since music always existed

And scales were known also but the association of a sound with frequency is the beginning of acoustics and developed in a very strong fashion it developed mostly by procedure which I'm going 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 the perfect a science must find in one aspect of

the mass 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 stick the applying and at this point in time much of the effort of many scientists still is not natural enough devoted to developing those senses which we are described and also of course to link them for example unified field Theories proposed to link the things are heavy and the feeling of light light and weight but on in all this extraordinary effort one sensor has become totally deprived

of implementation science and that's a sense of roughness the the problem was very very far back being very much one of those knots who read all books I was very impressed to find in Plato one of his dialogues least her senses and what I just said the scientists Corresponding to it and then he says roughness is well didn't say much about it and roughness indeed turns out to be far more complicated than the other the other sciences to require mathematics which was a market higher markedly higher level of complication on end of difficulty so what

my my whole life has aimed at was to find the regularities in variances in Sciences all is only study with variances in variances in our experience of roughness which might Apply throughout and might provide tools for attacking roughness in all its aspects so when when I say that fractals which is the world that corn for a certain kind of orderly roughness they occur in mathematics I will give you at least one example the sciences I'll give you few examples in passing engineering in finance I'll spend some time in finance now sometimes one wonders why all

these days topics are brought together they Are not brought together for any fundamental reason except that they all exemplify once again this very simple environmental issue the question of roughness so as I see on first line fractals are simple complex and open-ended what I mean by that if one begins to define fractals without the kind of cloud of sophistication regular and mathematical elusiveness which was given to them for quite a while when they were purely part of mathematics if One forgets about these complications actually simple in fact children to a taught the ideas very very

early on but these very simple rules create extraordinary 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 to cover them now everybody knows me for pictures and for Some pictures who has quite the 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 in the light could be have been tamed for the purpose so different from

its original purpose but my experience is that if I start throwing pictures I will never end the great fun for you for me but it's out of place so I will I was simply not not dwell on them I will just show you this one well let's now get from my point what is roughness and how to how to handle it well it's very appropriate that this auditorium is a building of Earth Sciences because nurse scientists and by the Miss wick me trough gave me this picture he told me that when he trust geology to

his students he is very careful to tell Them at every given time always put an amount of scale because if you don't put them in the scale you'll forget how big the thing was after Russia photographed so you put yourself a 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 example indeed of this confusion this was cheating is very easy to cheat because indeed this type of terrain is the same at all scales now to Be

a same at all scales is at this level an element of purely fault floor and I will hasten to go ahead and speak of Finance since I spent very many meters of my life both in the 60s and again recently studying financial prices in finance there is also a piece of folklore which says that if you put price a price graph somewhere be careful to put a scale of time because otherwise one would not know whether the price changes or prices have every minute Every second well that's recent already they are every year it's all

the same goes up north wing zigzags and so on this is for florrum and I'm not on a shame are embarrassed to speak of folklore because in a field like optics folklore has been how to say taken advantage of 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 Fourth law can be can be how to say change modified to fit other purposes now what is this Walter about primarily now this edible vegetable indicates its structure it is

cauliflower obviously its most striking feature as everyone knows that is that one takes a florid as opposed to whole cauliflower and drops everything else the florid looks exactly like a small cauliflower and then I do it again again several times well it's very interesting that this this feature Was known to everyone but I found very little mention of it until my work perhaps the the cauliflowers are very interesting they look not from side the very top and there are all kinds of Fibonacci series all kinds of structures in them these attract the very much attention

but I'm not interested here in in how to say precise description of the arrangement of Flores on something much much more important which is the roughness certainly surface of this this Vegetables very rough is thereof in a very excessive fashion that is highest for the hierarchy a big thing is made of parts identical the 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 the this idea of large and small-scale parts is of course all mixed up and confuse them now

this marvelous graphic is due to my friend Richard was to implement a model of Mountains that I put forward in the way back in the early seventies before computer was able to do anything of this sort now in it it is not a picture photograph it is not 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 was very much attacked on his account the idea is simply that there is an invariance Between the roughness of mountains

in big small and in very small scales now again it was not something which I had to say quote unquote invented explorers of the mountains whimper who climbed the one Matterhorn at least was the first man recorded a slang Matterhorn wrote about his his rambles in the Alps that a small part of the of the landscape is same as a big part and he had all kinds of philosophical consequences from it but It was certain not taken seriously so what I did here it was just to embody this idea of wisdom this wisdom of folklore

and replaced by invariance now one knows these variances which occur in in activity see quantum mechanics and so on hidden variances that those of dilation and reduction if you dilated and reduce it appropriately it doesn't change well we can go on forever and but I would like first of all to give an idea of the natural Problem and just how it how always past the the idea of fatality I flashed on the world that coined in 1975 the idea of last allottee did not exist before but an intuition of fatality and in fact examples of

use of rest ality go back absolutely to time immemorial this picture here is at a village in Tanzania of a nation which I don't think lives there it's a ruin but if one looks at this that feels carefully one looks mostly for a top which is less nice Photographer but more telling one realizes that's made entirely in a hardcore fashion there is a whole village here that this distinguish 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 the Dakar device was known again to every country I chose Africa I might have chosen just as well temples in many From parts of the world now we get to to a point or two points rather to the right of this picture you see the celebrated face of war of Dali I don't think that in you about returns to the left you see this abstraction which has a long history it is actually I saw it for I first discovered myself discovered quote unquote invented world

whole world and then I realized that a man named shape is he had written a paper about it and I need a name a call Shabiha gasket to make it sort of nice and it is had become very important shape in physics you see exactly how it's built I don't know I have to tell you take a triangle you take away small triangles now the the most astounding factory that you go to 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 in decorative Design so to summarize and fractals have on the one hand a very very old history and then this shape the triangle was was drawn first around 1900 then was a period in which mathematicians started distilling them did they know about the uses of fractals in the decoration I don't know there is no record of the talent however tantor who was one of the great among them did write the record in Writing of his thinking and he certainly wrote to his friend Dedekind repeatedly that we mathematician

are of a divine race and we can devise shapes that nature does not know so at least one of them thought they were inventing something in fact they're not they were putting in a very very precise form some structures which humanity has been always familiar with and so then the question arises what if you want to begin a science you certainly must need Numbers for the for it and the astounding finding was the case at least 15 years ago that whereas of course loudness had been measured very well which has measured very well by a

frequency and the same thing for for visual impressions weight has been measured temperature had been measured that was actually invention in history by Galileo there was no measurement of roughness and I wrote in mid-eighties nautical With some metallurgist friend about roughness and he showed me after after many examples that in the literature of broken metals of fractures or metals the all kinds of measurements of roughness are very very imperfect and very very unsatisfactory well so I introduced in science a notion which which I call fractal dimension which I 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 I'd how's the dimension is impossible to defeat in science it has no there's no way to measure it because operations it embodies the other dimensions are indeed immeasurable and in fact something very interesting happened you will see these dimensions come in fractions and after several years of working with them in fact doing nothing else except using them in one context or another I I Didn't see I taught myself about about

about roughness because everybody I think not about that I taught myself a correspondence between roughness and and this number an anecdote at one time a friend came to visit me at 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 once in 1.2 and Yoshie 1.8 just out of the hats and I said this a little bit short of 1.5 it was 1.48 And then I'll come back to an example of what well guess dimensions for tourism because of being

attuned to it again I don't think that I have a skill which is unique I have established myself a correspondence between this number now how does number come it starts with a very very trivial property of dimension ordinary cases you divide a string into n parts or a square hand 15m parts in each case dimension is the ratio of log of n divided by log 1 Over R whenever I publish that a scientific journal and idiot editor tell me which base of course doesn't matter because the ratio now you are putting the the usual usual

shapes you take em an interval you replaced by the zigzag well for parts and you repeat again again well here if you take the log n over log 1 over R you get 1.26 etc in the middle part you get 1.5 exactly in the right path you may we get point to 2.0 I'm not going to go into details of It it would take forever however this and related quantities have been totally tamed to measure that we measured sometimes with exquisite precision in one object which I to which my current latest paper is devoted the

dimension is 1.71 with some uncertainty about the next decimal it's well uncertain not much but it's astounding that the notion which is abstract when properly generalized and so on becomes so completely measurable and so let me Now bring this matter of for tourism now this picture is my book I forgot which page I was on I was very much in this brown motion Brownian motion is one of the center of probability theory of course it's a process which was defined by Norbert Wiener and 20s and was his greatest claim to fame for a long time

it's a drunkards walk and you see this black black line goes around it so I was drawing these things all the time and in a certain sense I was fishing Well maybe fishing well I'm very strong impression that my collision friends just underestimate the power of their eye well there's something else to it people are more or less good the eye I mean it's very clear scientists are good mostly on exams which measure ability to do algebra fast and correctly so it's not necessarily case that scientists have very good eye at least not 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 playing and then decided that the whole this thing was a bit too diffuse and the Brawn emotion were starting here and going around around around ending here so everything was looking for was invisible everything else looking at had been known before so I decided first of all to have the snake by this tail by having ocean motion come back to where I started And that's again idea of Norbert Wiener it's called

a brown region and then I decided to color it color doesn't mean that put fancy colorful things like on these mountains on the Mandelbrot set it's just very simply to make a different beam inside and outside so all the points which could be gotten from outside on are white all the points which are not accessible I made are put in in gray when I saw this picture first my screen I tell you I've been Speechless the picture screamed at me I am an island a very complicated island now as ed has said the whole long

across of Britain and you can replace any island as you wish in it is a very basic issue and the length isn't like 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 and and and fake ones and this one was very very irregular so my first reaction was about The most difficult islands I've seen were foresters 1.5 and got beyond 4/3 was a magic number for third world was the islands I had I was very much pleased with that being sort

of irregular then I measured it because my can measure it and the so was that the dimension was one point three three three three six well this is a humongous simulation so you can get these things very accurately it was for tourism now the most extraordinary thing About that is that it begins 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 pain which have been sort of made bigger that England of course had been vanished before different

Island Scandinavia was there a bit out of work and people begin to see actual geographical features and very easily and that's sort of almost o matic you always do curves of this dimension and You find it well for thirds was for a long time a conjecture for 18 years then a friend of mine build up the 20 plot he gave a demonstration as a 4/3 which was very beautiful but not rigorous he calls that exact but not rigorous it uses all techniques of physics which are probably correct but are not certified as such and then

shortly afterwards SRAM Lawler and and the vendor proved it now what is interesting here is not that the theorem was proven actually proof take several Hundred pages is extremely difficult it the techniques of every kind is something overwhelming thing the interesting thing was the extraordinary short path between a childishly simple thing namely a drunkard a drunkard going around and then a question which is very difficult but for 20 years baffled baffled the most brilliant among my friends and all the topic of great despair first of all they said I'm going to do overnight then next

week next next Month next year finally to the Millennium well didn't take a millennium but it had to wait on the next millennium it's very often the case and I think that actually it is representative of a theorem or what I say about roughness that roughness is have not been explored systematically for itself I mean I made some devotion to it but was not there for it is not a case in which simple Problems have been solved long ago and only difficult ones remains simple to state I mean it's also a field a field which

is not very old so there very few cases in which a prime arose out of previous problem problem is the question mathematical questions arise out of things which are very very common and I will tell you at the end more about my very great commitment now to education and one of the aspects education is to him is to realize that Even adolescence sometimes the small children high junior high and so on are very comfortable with the concept of that describe roughness the concept of self-similarity concept of algorism i mean if if no affiliation is added

they understand very well and then when when they see that by a little manipulation from these very very simple things when gages very complicate matters I think that we make them understand to at least some extent this notion of extraordinary Fruitfulness of science entry in in creating primes as it develops and well it's a course at Yale which I expressed very hard with micro framers tell you more about that it's the last line token of the value of the eyes recently turned to pure mathematics expression of that is that this neutral about that in a

French tradition of setting American and the illustrator felt obliged to mark the boundary of the subject the whole point that the boundary was not didn't have Been marked the boundary was visible without the impact you didn't have to mark it well the the all that cries out for for simpler proofs and I'm going to to to go further the critical application clusters I would not go into into the case of the oddities but the general idea is is very close to what the structural magnet isn't a magnet is made of lead magnets going up or

down and there are critical percolation clusters Complicated 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 third fourth it was an object of great fascination measured after I I brought out 4/3 and again people on ship did something about it and and Stas Smirnoff proved it earlier this year it's a it's a whole field which as a reason which no is completely independent independent of the eye of course but which would not Have arisen without the eye and I really think that's

very important from viewpoint of an understanding of science to to realize that the what has happened the reason why the I went out of of the hard sciences was not hots a decision by some committee to neglect it it was observation that last time time went on a new questions did not come from the I new questions came from all questions also the fact that once those Sciences were established again The simplest things were very rapidly done and then that more complicated things came from simpler things in the field itself but in in the case

of the prices of the roughness and of the topics are discussed all my life she's very different are still very very close to to the initial basic issues and therefore the I had been absolutely indispensable the I and sometimes even 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 very fact very very different this sounds very different the 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 the when I became fascinated by the subject discovered and described it in in around twenty nine eighty I started with the definition which is Different from the one which everybody now uses and it turned out that very difficult to to do numerically on the computer at that time very very clumsy and so I changed definition which is a process map additions like but in this case I'll show you definition thinking are two definitions identical and the second definition is one everybody knows is very

easy to put a program and all these incredible riches came out pouring but And observations made on the pictures became mathematical conjectures and the modified proven great success but the initial idea that the two definitions are identical well one set is the closure of the other that assumption conjecture whatever is still not proven we use that and brand brownian cluster in this course at Yale on Infractions phenomena titian and also in workshops for high school teachers because they there is nothing there which has to be Left on explain everything is then we'll 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 it 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 with price financial crisis in the early sixties And then topic which I very much devote very big part of my life to at the press at present now prices of course go up and down that was not

to everybody and all kinds of nice maximum is about it in 1900 an incredible genius local problem his name was liberally nobody notice him he had a very miserable life but he wrote a 900 miles believe it or not called the seer speculation speculation meant a speculation on the stock market or bond market and the heater used for The first time in loose and incomplete fashion the Bryan motion which Weiner later made into a central mathematical topic and which means in between Einstein and others made the central physical topic now the idea of bascially was

more or less that prices vary at random you can't predict them you toss a coin if it's heads you price blow up its tails price goes down and you go on and on and on well you can approximate that by sequence of Independent variables and you sit on top very clearly the sequence of independent Ossian variables which are the increments of Brownian motion it's also quite noisy realistic all white noises we call gaussian Gaussian white noise it makes a very big difference I'm going to argue in a second now a much later a whole theory

of stock market occurred on the basis that this model bascially is indeed representation of reality and the very the size of these increments of Price the this size is what's called volatility the model assumes constant volatility and indeed if once you're seen a little piece of this line on top you have seen them all don't very much alike and if you average it you take the averages moving averages this whole thing goes to 0 very very quickly now look immediately at the bottom five lines skipped or come back to them later at least one of

them is a real price serious price changes or at least one of Them is illustration of my current forgery which is called the multi fractional model of price variation now I'm sure that many many people here are such great geniuses that we are going to guess meat which is which but you are exceptional most people can do it because all these all these phenomena have a very strange fatalistic that a very big Peaks all the time and the peaks don't arise by themselves the rise in the middle of periods of rate Volatility then the peers

of a very low volatility the appearance of reduce oil 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 be impossible to grasp and so how are we going to represent those phenomena and I will put this this example in particular because it puts in contrast very very sharply the approach to roughness which which is Had 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 non calculations then the approach was more people have been advocating is if a constant volatility phenomenon doesn't work then replace it by something which is a variable relativity and if that doesn't work been replaced by something at discontinuities or jumps well this is very much the Spirit of the celebrated model of the motion of planets due to polymers in the Ptolemaic model the planets grow around circles and the Epis a epicycle cycles which are and then the epicycles plants growing on circles which centre of which grows another circle and

then in addition one must assume that the center of the basic circle is not identical to the centre of the universe in fact the C factors everybody knows that how kepler replaced that by a Different formula which later Newton transformed by explaining it through gravitation but the main fact is that that two approaches to to this phenomenon either that the prices which are in the lower part are just other examples of something or something complicated and pleasant which are just combinations of the simple all that there is something radically new which involved now tell you

about how radically new this Has to be many of these these very big Peaks you see on the five the lower lines exceed well something which is defined a standard deviation actually I don't know how to exactly measure but sort of ten times bigger than the standard deviation of the most examples that called ten Sigma events now ten Sigma events in the Gaussian process Gaussian indepent process on top the probability will be 1 million millions and millions and millions the inverse of A picados number it's a very small number if that were the case these

things will never happen but as you see here to happen all the time so it's not a matter of adding a little bit of large values because one good situation in which the large values dominate everything very strongly as a matter of fact I'm going to go a bit illogically to make make a point early on the central aspect of statistical physics that is Roselyn AMEX is that the big System evolves in a certain way namely entropy increases with probability one the probability zero entropy is going to decrease but the events of probability zero are

totally totally negligible the events which count are those of probability 1 a average of course converging everything the past the future independent all kinds of beautiful theorems apply but if you know to to to financial prices it's again part of folklore if you look at last ten Years and the wonder assuming that somebody with miraculous powers of of seeing the dark some devil would pick the days that matter you realize that that the desert matter over just ten days out of last year's very few days mattered and the great fortunes were made in very few

days great ruins happened very few days so one gets ultimately decision which is very very unsettling that is in this context only the very few rare events Account overwhelmingly and the rest doesn't count hard thoughts hardly at all it's a way in which context which one must review review ones and ones ones the intuition very strongly but reviewing not reviewing it's best not abandon solid ground and abandoned the principle of modeling the primary principle of modeling which is that of invariance it becomes almost a joke Isis before to others to extend some Scientists and one

of them emphasizes that the basic three society's variance you finds the right invariance and in fundamental physics you look for all kinds of new invariant system and the lucky ones or the perspective equals ones are and the ones who find better theories in this case the matter was to find the right invariance and so I'm giving you them the key the the forgeries which I'm which I introduced in among these these lines were not Chosen by taking the line on top and adding this adding diet adding listening that until the thing looks more or less

reasonable it can be done but I don't do this way the the key is to find right invariance and lo and behold the right invariance gives this result now let me identic step back up because it's a lesson which no no not only applies to this financial crisis but to 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 is no correlation but I believe it is full of very interesting structures having a white noise to be so interesting I'm

used to white noise like on top which is boring which is like you've seen one red would have seen them all like former President Reagan said the bottom everything is different well the fact is the bottom five lines are not Gaussian therefore not being if not Gaussian whiteness does not imply Boredom whiteness implies a very very often a very rough ride them now the the second ID so the two of them a tool of science namely the spectrum which had been marvelous is not applicable this context it doesn't work I mean again Wiener was very

clear in when he described spectral analysis early early on as you realize winner one of my intellectual heroes and the he want his rabbit the conditional which spectra are good but these conditions Are not satisfied for this for this data now is that a new finding by big Mandelbrot that these are white by no means I remember when spectrum B team became usable after the fast Fourier transform was discovered rediscovered rediscovered actually everybody went on to specialize everything in sight and prices were sterilized the Eternity wide price changes and the result was absolute absolute horror

how can I be if it's White should be like the on top now every book of per mathematic says that independence and orthogonality are the same for Gaussian processes orthogonality means a wide 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 of these examples the examples made by culture in so far as the stock market is not part of nature therefore as part of culture and or made By me which is also cultural but so you can have structures

in this fashion now the only thing is the habit came in in from thermodynamics that if you take a big enough system then fluctuation does not matter in fact the whole single proving the larger bigger and bigger system ultimately fluctuations both matter and the average out but look at the single left it is it is not quite the same thing all right it is squares of it and there's some hanky-panky to it But it's about roughly speaking variation of prices over a century if this these real things the right were just fleeting defects which averaged

out then the over or a century have 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 a self of an invariant processes the same a small scale a big scale self similarity or variants Thereof enter into this game and so let you know it so the the models I'm going to say a few more words about that most fractal models but the second layer on top was my first model of prices it I called

meso fractal now it's a new term because I need these terms for various reasons at that time I was so focused on large values that I've neglected the dependence between prices the bottom line is bottom lines show you the price of enormous dependence I mean That's why people can give you a prediction actually possible and well anyone from another but that's not different issue of here but the the second line was independence have these very long Peaks and that took care of one other problems which is the long-term dependence the long tails the large values

then I had another way the second line take care of long-term dependence but only multifractal took care of both it's interesting that multi Fractions but the fact allottee is a notion which I introduced in about the time I met Ed Laurence I'm at first a little bit later to explain the nature of of dissipation of turbulence not explain them so I take it back of describe it explained 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 self Sameness which is no longer search similarity but self affinity enters around so this is not a model of price change it's a cartoon what's the cartoon a cartoon is something which has which has recognizably same features but which is simpler to draw and to behold so on top you have a straight line a trend which the price changed from beginning to end I was told by all means it must not be going up so it's going up And then there is this this variation of this price which

is second line which I call generator goes up up down up and to simplify it I make it symmetric then for each of next stages I do is to take this generator and squeeze it but you see my squeeze it in different ratios this way in that way if the ratio of squeezing were the same both direction you would deal with a similarity and the Signum is self-similar the switch at different Rate rate so this one this way it is an affinity mathematically therefore these things are self a fine and if you goes down you

turn the thing back it was again again now after a while you get this jagged curve which has certain amount of boring regularity so you inject the minimum amount of randomness and to the song you simply at least at each before you do you introduce generator you choose at random between see possibilities down up up up down up Or up up down a very easy to simulate and then here is what you get was different on top with the values I you saw in previous picture you get the sequence which is more or less what

Ashley was saying more or less what the conventional so-called feared market supposes coin-tossing or a variant there are then what you do is to come back to precinct you move the top the stop point a little bit to left so you just you have one parameter here you see one one Point parameter the point parameter is the first break determines the generator and some as you do it you realize that suddenly this thing becomes a memory regular now this order is to it it is not an incredible accumulation of cycles epicycles not central circle cetera

cetera it's simply cell affinity this these curves are increments settling process which is certifying the construction because you do it by this Recursive process which has the minimum amount of of randomness name is shuffling and at the end you get something which is none of the realistic for stock market because it seldom happens that you have experience but almost nothing happens if you enter in further you got even further so you can go beyond what you observe stock market now I don't see the model I just say that it's a case in which the first

investment gives a high yield and I have Been many sciences and I'm also history not solid records of many early days of many many successful theories in science and there is a 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 just overburdened to begin With them now I don't say this one is

going to is going to take off it's very widely respected very widely criticized is very widely hated it's very widely etc etc but it the point is that it provides what I think was the answer in this context to have an extremely short no bells no whistles path between folklore and a quantitative measurable realistic and also realistic representation of data but now here is something which is very disquieting so You see on top of these things zigzag is not the same before but again you just take the exact and you do it recursively by changing

the sequence each time and so these eight sequences of increments are identical from viewpoint a process generates them and here we get into one of the most constant and the most in fact perturbing features of fractal phenomena and the perhaps how I would I would first I would first comment on that If you apply to these eight and eight processes the usual techniques which applied to two random very very phenomenon you find that they're all different the variance is different the correlation is different well in this case not even white because I didn't make sure

the white it's very complicated if you apply the right techniques which are techniques of multifractal analysis which developed for the purpose of measuring roughness Then all these eight are the same let me come back now to this matter of the author of metals I mentioned earlier so dumped a soldier who's metallurgist was getting from whoever national your standards or maybe it's already different names sort of finger shaped pieces of metal in fact the team in large number so did the National Bureau took a piece of metal and had an extraordinary careful treatment with the expensive

piece of metal they are very Carefully treated at a constant temperature or for a long time so D homogeneous and then they are cut into five pieces and sent to different laboratories order them we caught them bang bang bang bang we measured the roughness as as the book of metallurgy said it should be done and that measurement was low and behold exactly the same as the measurement advocate in the books of Finance that is ute approximate this method by a Straight line or by playing or whatever and took at the root-mean-square deviation from that level

and you assumed root mean square deviation is a measure of non flatness well it is a measure of not non flatness but the horrible fact was that we had five samples which the National Bureau standards guaranteed to be identical and the the measures of roughness were all over the all over the landscape so didn't measure anything if you measure The root squaring these things get results for a completely all over landscape these things are totally different the suggest next round level very variability however is then measured for the for the for the the these fractures

when you measured the flatten 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 For surgeon well not quite 4/3 forces for lines and it's a normal dimensional line is one curve so two plus one-third and for surfaces two plus one-third some the and then people redid the experimental very much longer samples and they found that indeed the 4/3 holds over five orders of magnitude that is here is the 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 landscapes landscape by looking at the right measurement you find the certain values and measured over five or more order of magnitude the heat treatment only changes the cross overs at the end and makes this range bigger or smaller but the precision of this of that is absolutely incredible it's also true for for glasses is true for variety of Extron the different surfaces rough surfaces have this property of plus One-third now I

view that as being first of all an example of a mixture of nature reduced to a manageable concise and clean conjecture it is not a completely statement of structural metals a simple statement that thing is for surgeon and it is of course the greatest clear successor latest token or successor for a model which say that this way of measuring it now after having given you this example with Construction I want to describe this in physics it's very important once you have a certain model to change some some characteristics temperature and something else anti phase diagram

in which one sees how the property is a system vary and so there are three ways of showing it for the symmetric again going back to prices for symmetrically generators and and well let me let me show this and I come back to the previous thing this this is them on the Left on the right a test which tests whether price changes are indeed unit followed the multi-platinum model here the price in the question is is the exchange rate between the dollar and Deutschmark which had the advantage of being rather clean finite samples since on

January first it was top since the Dutch mark will vanish but it was 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 of Finance the data act in physics in the best cases someone measured when does the diagram and the towel 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 there is this envelope of these lines and

this envelope is called f of alpha and these things are the the the fundamental characteristics of of The processes so to come back to to these if one did the same analysis of these on these eight you find the same results again every analysis written appropriate to find very different results so here is the this matter of of manichaean view of the world when I describe to you in the beginning that roughly smooth of course I have mind the fact that that item that even though smoothness is so rare in in our experience it was

a transformed into Geometry in the marvelous fashion and now was myself as a young man totally immersed in classical geometry the most wonderful thing one could imagine and it by by this kind of extraordinary miracle it gave rise to a description of nature beginning by ellipses being good for teller for describing planets motion etc etc the whole of science is dominated by mathematics which is that was smooth and that mathematics arose early on derivatives of course are are part of it Very strongly and and most differentiable manifolds and so on you find his worth differentiable

all the time these are the primary objects of mathematics and particular of Science in gravitation and light but now we have that decision that in some parts of nature things don't fit in this mold at all so by this belated process process which started with very high 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 feed themselves from nature by trying to feed themselves from nature they introduced these concepts and they related related ideas which were called mathematical monsters and

the like to Ankara was many making fun of of Cantor and piano by saying that in the past partitions were inventing new shapes to help us understand nature but Today they invent new shapes just to annoy me well it was the case and so the the central point of actual geometry is that actually a counter was just plain dead wrong that these shapes were not the way no invented is irrelevant whether he knew of the correct the various schemes are not as relevant that this mathematics was actually necessary to take the first step for study

of roughness and so that brings too bad but again into in very strong Relief this question of averaging which destroys randomness the classical pattern was that of thermodynamics I don't know what anybody teaches that any longer or classical thermodynamics randomness it exists and then there was a statistical thermodynamics in which energy was not a constant but was fluctuating a little bit but so little only the fine electronic measurements could pick the thermal noise and that was discovered only rather recently Namely about time I was born for everybody that this is the finish of recently but

but then the phenomena we deal here are not like that things don't a result and so what I have developed increasingly over the years gradually but focused not very strongly is again a view which I would love to give you Morris to finish this talk of states of randomness now in mechanics we all reveal the laws of mechanics are unique well forget about details for the Phenomena we're looking at it's all classical mechanics Newton and so on laws are unique however if you look at the assembly of molecules it matters very much whether the temperature

and pressure are such that the whole thing is a liquid gas or a crystal or a solid dozen we used to say it matters very greatly so one does not begin by starting everything I'd initial on first look it's a gas I know which tools to use it's liquid I know it's tools to 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 to emphasize that there is a beautiful axiomatic good for everybody but then once you go to practice especially

practice in the physics and engineering and finance one always looked for a phenomenon which averages everything sort of averaged out two limits very quickly Was neither rhombus after a certain time but the phenomena which which I study whether it is in nature and that means turbulence or whatever or in culture that is stock market or everywhere this phenomena do not give any evidence of being averaging I call them I call them widely random and and the the very simple formulas in the context give very very 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 at actin by my friends mostly my friends enemies did not even bother because i describing an alternative view of nature which was so different from the view based on smoothness smoothness in whose include of course everything which is ruled by partial differential equations Laplace full hip was know that he stalks and this thing had not the French not derivatives had strange behavior and so On so do these things belong to different universes well I

don't think me do I think it's just very unified I think that that partial differential equations if pushed beyond the conditions under which the stringent conditions under which solutions are unique and smooth and differentiable and everything they give rise to very complicated behavior and one example of it is that we take a large assembly of particles interacting by inverse square Root attraction a la Newton if you start with Poisson distribution or just or just the lattice and put some pinch of energy to start it after a very short time you see that these these particles

go because strongly complicate and reassemble themselves into a fractal universe I didn't mention factory inverse until now but I would like to see two words about it distribution of galaxies the universe is Factor there is no density all kinds of things occur which are quite quite new well that would suggest the fact that the clustering of galaxies could very well not be due to specific specific complicate forces which we don't know but the simple behavior of particles under one of our square attraction and so that rate equation is clustering up there in the sky or

is up there in the head well it may well be in her head I don't know but it's the question which Has been discussed in fact one when finds that many of these questions become very very much revised now from begins to him to include the flat fan of the universe oh yes it's a fact early on which is sort of twenty years ago when I wrote my book universe was prospectively factor up to five megaparsecs fifteen million light years depth is nothing by universal standards but then except to 20 to 50 to 100 I

don't know is between two and three Hundred for most people now and many people believe it's one thousand which belief our so if it is true that the universe rational up to that level then fractality will insert itself into cosmology in a very strong fashion therefore not only fact early to be a result of very simple laws of physics like Roswell was possibly but effect understanding about your limit Rosen well I must stop here because since I'm A cop more complicated become very complicated and it's not for this lecture I've been writing on this topic

for a long time as you see and also for those scientists in finance the journal quantity finance has several papers in the current year on that I publish right left that's my book on finance my book on or noise I would have to end by this one which is coming out in a few weeks or a few days perhaps by frame myself on mathematics education we find to our Utter amazement that humans like fractals so and I won't mention him at art but this was a result of a very very very poor programmers work he

had some bug a semicolon or something not dreadful it gives the result he stopped apologizing feeling to be filed I started thinking from what on my heart and before correcting the the the bug we had his picture taken it's something on the mandible said Mistake was made but it's a whole topic unto itself that that humans seem to be happy with roughness many you must have say experience as I do because many that you say I'm a petition of 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 mechanics of the flat and the smooth the mathematics of the Rafah has developed by many people hundred years ago and now now it's galloping development on all Sides never gets this reaction fantasy if I had only done that in high school I would have loved mathematics well you don't need to be told that math next lovable but it's M it's 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 the first good and bad

the evil you choose which one which smooth and rough but I think it has illuminated my view of how things occurred and the reason why roughness Came so so late why so difficult is because it simply is more difficult it requires DIF mathematics which was developed on a recently which 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 Thank You professor mantle brought 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 [Music] much more than you think and much more than I thought not so not so long ago and we have now for several years that here in at Yale a workshop during summer For high school teacher so we get the cream of Connecticut so they are the best but they tell us

stories which are quite incredible and they asks about how electronic crime can bring it is your 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 and some of friends have tried with the kindergarten children they had they put them blocks the blocks Had o or I or X and still play with that then you put other fractal things and the challenge is to rebuild the fraction by putting this in certain order the case becomes so absorbed apparently that

they are lost in because that's the real task it is not a task which did you that teacher has imposed it tests that nature has imposed on humanity for forever and forever and so the in junior high is very easy everybody who is anybody knows how to work reform at eleven which put Me to shame because I never did learn to program but so it 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 the mathematics was smooth had been

so elaborated I had become very very far from experience and the questions it us are so refined that it's that it is it is not question which come naturally the not questions about them about anything on beam has thought the best Concern about the about the circles I was a geometry not I love that but I was the only one who loved it for all the other it was artificial thing that somebody mean Pascal just made to bother them but question about about about mastering roughness which after again they rule and not the exception smoothness

is engineering stable smooth well not quite supposed to smooth because it's only several years there so it has some roughness I'm sure Fracture but on an approximation smooth it's engineering which scold which is dry and cetera et cetera so so if mathematics can kind can go back back to its origins and it is not even something which is artificial for example I mean give an example while I'm discussing that with a friend day I introduced a notion which is which was taken as being the end of madness that of negative dimension now you measure things

so I measure roughness and now you can also Measure vacuity emptiness so listen to that if you take two lines in 300-page space their intersection well they're not the same intersect of course but intersection of lines is less empty than the exceptional line the 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 long story but then to explain it we must understand that that lines are not lines the lines a

little tubes and lines being little tubes is not only invention of me estimation of minkovski was very great man 1900 and who point out that when this away from all kinds of horrible paradoxes in mathematics by never thinking of lines on lines or surfaces surface but thinking of of lines as being tube surface edenia become fortresses realize cetera so the When goes back to these things not only when one goes back to mother earth to get strength again but when does in the fashion of fruitful because without this element of going back into epsilon neighborhoods

to call him by a fancy tail term one could not make sense out negative dimension and negative dimension is something you can measure in which a useful thing in the study of of turbulence we have another question well certainly certainly short market The human behavior as a matter of fact I have come to use the word culture in the sense which is perhaps not not smooth to some some ears I speak the fractal geometry of nature and then giving an example like stock market but taken take example of an internet it's human behavior it's not

exactly what you think human behavior but large number of very billion people working on cross purposes and put this thing together and works Marvel's most a time and every so often It's terrible and your messages don't go through but so people first tried to apply to the internet the techniques which had worked for for telephones that was placed on West's own behavior and so on it was absolutely off the mark it is a multi-platform and it has to be multi-platform now after the fact is understood so here is the human behavior and it was design

of this design very informal and noisy and messy design of this huge system which for reasons which Are after the fact so the vaguely understood is multifractal but you don't have to understand it to live with it because you have to live with it and so if you if you make me equipment you better test whether its properties will be his well in the fact and face of it but I'm trying I was try to to speak 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 the mind-boggling my friend my friend I should Russian had found experimentally that that music is one of ref noise like these things I was showing you here he's a physicist and he is not a loud person so it was sort of didn't become very widely known but then I was approached independently by two by two composers one in New York and one in Europe and you must know the names the charles warren and george Ligety who

Told me that looking at my pictures made them understand nature of the craft and as for each and they said well I speaking in voice of either because they exactly same same explanation I have been trained in a very classical conservatory traditions and you very well about all the instruments their values it's about everything but I was not told one basic thing which is what distinguishes a music piece of music from a collection Of noises and I learned they said each of them said by trial and error I brought the composition to my teacher said

to busy and I brought another composition it's not decorate enough and another proposition it just goes up and down too much doesn't go up but not too much after a while I finally understood what to do and I've been doing that very well and I'm sure it in or Det and worgen there are very famous people and the pictures it's obvious it's obvious Except nobody told me that but if a stone that is a sonata 21 Minister three movements Allegro lento presto different each movement loud soft very loud each so the thing must have structured

on scales something will be changing at every scale and that's it if you get that you get at least bad music and bad music is much better noise and everybody sing by music and voice was telling a story about about dissipate atonal music and Chinese say this bad Korean music Korean bad Japanese music Japanese bad time music whatever it but was music of some barbarian race but so that the essence is not at all fundamental again wouldn't bad musical together no sorry I don't know unknown in painting so in classical in the classical landscape a

very artificial form of art both in oriental and European tradition so it was big tree which framed it and then this little man and so on all these this Rules of composition if you look at it one big thing and most of the books of art did speak of of design and I tell you what I did discover I do my own my own covers for my books this channel well I like to do it so so the this this book for teaching you has every single cliche of proposition because I felt it would be

effective so but on the other hand I was a look at the film of Kandinsky painting all his paintings it looked exactly like a Program of do fractals with joke she would look at this piece of a thing and then bung a big line solid line it's a it's you see that inaudible then.who small lines on the film ends has small small small brush and he's adding little things all over the place so he puts big small and medium strong elements of scale which is exactly the kind of basic bottom motif like a fatality I

could go on forever and forever so the fact that that is DISA Speaking of current behavior that but but if you look at at at this African village some this behaviors not not in the sense of a physiologist but the sense of the artist on the ground now physiology system that's not imagined now I'm many my friends tell me that they have a nervous recording look like these things I showed I be very much interest in seeing them and but I know very well that and if you describe some some surfaces in the body that

is Certainly a human behavior some surfaces are meant to be as small as possible given their volume and other as big as possible the skin should be smooth and out like a young child's skin and the longer inside must be as confused as complete as possible so the the two the two criteria of design lead to either smooth shapes or two extremely convoluted shapes which happened to have fractal fractal features the long for human long for example is branching a Fractal on twenty three levels twenty three levels is very healthy bifurcations very healthy number it's

a million I think we're going to need to wrap up right now but I'll invite all of you who want to stage to come on down and speak with Professor Mandelbrot for a few minutes and I'd like to conclude by thanking professor Lorentz for joining us today and let's give thanks to Professor model drug [Applause]