The Nature of Roughness in Mathematics, Science and Art

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The Nature of Roughness in Mathematics, Science and Art
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Special Lecture of Benoit Mandelbrot, ICM 2006.
Mathematics Computer animation Doubling the cube Different (Kate Ryan album) Natural number Multiplication sign Game theory Dressing (medical) Position operator Vector potential
Geometry Observational study Multiplication sign Student's t-test Mereology Rule of inference Event horizon Fraktalgeometrie Explosion Mathematics Goodness of fit Causality Spherical cap Natural number Different (Kate Ryan album) Queue (abstract data type) Circle Integer Antiderivative Monster group Euklidischer Ring Theory of relativity Forcing (mathematics) Algebraic structure Division (mathematics) Basis <Mathematik> Lattice (order) Equivalence relation Proof theory Radical (chemistry) Order (biology) Glattheit <Mathematik> Game theory Musical ensemble Directed graph Gradient descent Maß <Mathematik>
Statistical hypothesis testing Group action State of matter Multiplication sign Direction (geometry) Water vapor Mereology Order of magnitude Group representation Sign (mathematics) Coefficient of determination Mathematics Exclusive or Many-sorted logic Circle Descriptive statistics Area Nuclear space Curve Physicalism Measurement Substitute good Hand fan Category of being Triangle Figurate number Bounded variation Identical particles Sinc function Directed graph Spacetime Point (geometry) Slide rule Computer programming Observational study Divisor Point cloud Student's t-test Event horizon Fraktalgeometrie Frequency Natural number Term (mathematics) Modulform Energy level Liquid Acoustic shadow Set theory Condition number Motif (narrative) Time zone Surface Forcing (mathematics) Consistency Model theory Algebraic structure Basis <Mathematik> Total S.A. Line (geometry) Limit (category theory) Glattheit <Mathematik> Maß <Mathematik>
Point (geometry) Trail Functional (mathematics) Group action State of matter Multiplication sign Parameter (computer programming) Mereology Dimensional analysis Fraktalgeometrie Scheduling (computing) Power (physics) Product (business) Sign (mathematics) Many-sorted logic Different (Kate Ryan album) Natural number Modulform Körper <Algebra> Process (computing) Surface Expression Variance Line (geometry) Lattice (order) Measurement Numerical analysis Order (biology) Object (grammar)
Category of being Mathematics Physicalism Limit (category theory) Theory Directed graph Relief
Statistical hypothesis testing Greatest element Presentation of a group Group action State of matter Multiplication sign Modal logic Direction (geometry) 1 (number) Chaos (cosmogony) Water vapor Open set Parameter (computer programming) Mereology Proper map Mathematics Sign (mathematics) Casting (performing arts) Coefficient of determination Plane (geometry) Many-sorted logic Different (Kate Ryan album) Forest Series (mathematics) Monster group Descriptive statistics Physical system Covering space Process (computing) Generating set of a group Closed set Physicalism Special unitary group Measurement Hand fan Category of being Symmetry (physics) Order (biology) Weißes Rauschen Right angle Fundamental theorem of algebra Resultant Point (geometry) Computer programming Standard error Trail Functional (mathematics) Divisor Observational study Civil engineering Mass Graph coloring Event horizon Product (business) Power (physics) Element (mathematics) Frequency Thermodynamisches System Natural number Average Term (mathematics) Profil (magazine) Modulform Theorem Set theory Condition number Noise (electronics) Addition Weight Forcing (mathematics) Nichtkommutative Jordan-Algebra Model theory Mathematical analysis Algebraic structure Counting Total S.A. Basis <Mathematik> Incidence algebra Line (geometry) Brownian motion Numerical analysis Optics Game theory Musical ensemble
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What I would laugh today is 1st saying The meeting to invite me for inviting me for this lecture and then The present-day you some of Very many facets Was so high You should be for the extraordinary ride they presented and also the unit not Begin the cheers In that vital There often industry would now let us think is very different from the same time varied don't go when primitive man or woman was Looking at the world surrounding Hall many simple shades that if you once more artists and there Your 1 season Storm is strongly Lake this circle that he where it may be a few more 5 examples of 10 examples of where is of important various examples of everything Eds I was better off duty Excel the bar with everything was ready or not I hollered you lend be a hand good basis A division are often most anybody who writes The story knows that the story does not have much bite unless the babies Fight between good and evil or from duty in 1 way or another force of nature of this A 5th of the 5 . 6 coexistence between is more because this Somoza was taken very seriously in every culture of developed and then became geometry then became mathematics then begins But The basic fact is that Everything was Except for Perhaps fueled a chorus so what happened to the raft was intact and that of course not going to argue very shortly that There are often has a very strong relations too many artists From Time immemorial when sometimes people must be fooled as In fact of over fractals and my 1st sexual it's not me In fact it's nobody's under years of 200 years Somebody very very who 1st Made drawings which nature a simplified the leader of the you've eaten away all the traffic and there Official titles out from 3 shortly not have this stuff here of fractals numismatics it said Let me These boards 1 after the other not to as much of it Fractals had A very important role Our along the way 1900 the report monster but that Centerpointe there are not you would As EU them says of particular interest and when I came up in the 1970 yourself with a systematic study of them their people asked me whether how many interesting things for us practitioners are you bringing where I know of Way is 1 my virtues and the icy wait now It's quite extraordinary that the seat of the medals in recent years Have been given to people to people who pulled some conjectures of 1 is different and shows the force of the 4 Brown we ending their van pools with and And the Another 1 is different picture about Equivalent of 2 definitions of mandible said that which Heading for embassy and the cause of Monday when you order in large part for his work so humans 90 BC I believe the picture of the deterrent of a The introduction of the establishment of Duval old relations between the hand eye and the mind has been very fruitful Not proving anything because picture of prove anything but giving new ideas I to go natural sciences where you see the short Would what about pigs Still was a very simple nature of all very cool nature of the immediate nature and then cultural where the culture of it is sense that cultural just word for everything that Man has pleaded whether it's music of a painting the stock market being done not for ozone nature a phenomenon which might has created the but doesn't quite understand and still I would die For you The fact of the entering destructive and last but not least in fact forecast for most of the fact that the yuan is of every kind of an agent After the attack the radicals is not because I'm such a brilliant and not Uncle unskilled I thought I would be interested education event in my life where students to to me and say popular bottles Our and how much they them want to learn about mathematics so there was a door somewhere The door that nearly humanity to further This would end To finish my summary That is not surprising because you think I gain of primitive manner The world English My man has evolved and neural which left for the rule A smooth smoothness with the exception exception of particular simplicity in loneliness but know this look at this picture is very boring Borys just a piece of land and integer John descent of mine who took this picture of now he told me something a very important that He's School the professor was always tell students she role in and take pictures always end Betterment of steel so you'd know how big it is for example would put camera cap or put your said where the source look at that at a game where the picture was it was not a camera It wasn't drafted with with piece over a garbage bag what was what geologists
As a matter of released buddies by professor who students always put at the can be turned down Not as its creation of a compass or do it right But questioned why is why is it that you forget to put limit of each you Get from Peking mistaken and for fuel is fine of rocks with big So this is kind of a composer popular which I made a big use Of because It is not that almost every has some kind of 4 or distort somewhere and Case during the This is the same also and that is what practical not sought The 1st observer mission is That saves to various problems Mitchell in nature and culture I need to be Centeno but nobody could measures include the occult broke So now we can't do something Which is 1 of them a big Turning point of every signed a basic designed Our was All of us Cessation because otherwise how do we know the world with other were not because my tuition by Just started or by Do which died stone what which hear office but A series of light and sell at last we don't go out or eating hunger and so the court cholera Measure of the value Frequency This picture is measured by frequency as very simple temperatures measured by temperature of partners of need and the but for what office it was measurement of roughly wanted A few years ago Very strange old that was being done take The doors complicates landscape sectors sold any measure that really do that would a bit of good I think that this and the shade and 1 kind of tools for it and when a picture which shoulder rapidly It on to the next It's 1 of the very 1st Examples of of measurement all property of rough surfaces in this case fractures metals and he had some very recent were not so very much in which The data off A proper or a varied lot lot large changed 5 or 6 7 orders of magnitude which any signs He's a very very big and still need is idea The surfaces are ready and anger of the United States also has proven to be a key year end This study of abroad It is based upon that now so as signs of roughness If exist went do exist may called directly romantic on must find new examples of it I would not Do would you like stop at the last 1 Because of That important to Tyrona get through it and I'm very little time to speak about it it when I introduced a draft bills And And culture of stock market my friends say that but But a white with how funny living there of course can make use of it end Real found us so it may be a primitive But today everything would be stable it small its flat there The marble lobby of flat best slide and had been just for me well because source Of the regular shapes Had man who says he is ingenue mullahs but today People are students of mysterious summer students have no difficulty in inventing devices to sold problems and based upon the roughness now Profits was known for role in the vegetable world each of the features of cauliflower Cardiff no longer as of 9 for example when hazardous With the structure in which you think it because of the long period the structure of drinking Is the Please The very much saying because don't have not study using tractors with exquisite Precision to friends of mine his support by Paris and the Bible in back have run went beyond what the default because his knowledge basis for a description of the combined viable over measurements And in many other cities Which I brought the road and disadvantage of course Another test of a large dog the is the dollar demand have the same kind of love but Way is Done is to put drastic liquid plastic in the air The start of the lungs and then do which it becomes sort and depending how much you push you fill in more and more the of space now we had This model over A large dogs lung and a lot of it in our office and Mike of famous said at Yale and 1 of our distinguished for What we buy what is that That In the name space fatigue surface The other monster and sheer The licensee restrained very strange in terms of Roush of signs that would be Was evaluated Diane petition our own 18 90 defying the line that goes For every point of paying for or surfers of Ross who report space of these was something master which understand About nature of where that was used simply because they're looking at because these kind of structures were known for everybody world But motor Official but my friend of vast and the old that's included here the main ingredient more more is not a more precise figure is fine point of it but by
Set bit In some conditions very simple 1 right The area that from viewpoint of understanding of the shape of mountains of people had has been a very important step now O'Grady who was the 1st scientist time in a certain sense the broadest novelist text of which is that it The language of must motifs It is necessary to understand the nature and he stays and terms of triangles the circles and other directive figures without which It is humanly possible To understand A single word of the unit Book of nature when when they get you sit on the edge figure is quite obvious that he other geometric figures nuclear but also very clear that program that your start stuff Before reaching not that it would not distract among simple fashion at all and that his the What this Picture Represents the total since Oracle These artificial that it used to want a state since which is not in mathematics and physics in most sciences that achieved a certain level of development in a very instant you were the 1 times Reduced The rest Of reality to refusing don't change because everything changes but something exceptions North and Deaver the basic areas Now Question is Every draft of course If Every cell represented by scientists are of course the cells which can be be presented by by pure sine curves that are Duracell which killed a single fan can see in them Exclusive valves consonants are very complicated so the distinguishing as security zone which is not my speech Or even singing with consonants 1 has not speak Of pure lasted where you're fractals when you question is taking who invented fractals yes is impossible because they have been absolutely absolutely there For and for ever ask that as those Hughes example a photograph of African villages known the name of the village and they are unknown because it is not or in case was there before the These are all but different look carefully which is difficult at this photograph went fine that It was structure putative fashion with Volvo and Bender was Overall Most Although for East Haddam although for his chief minister Everything will be built upon all but this from closer and closer also was being in business This is scary picture of that It has shown the corals Of war and made after here is betting that in DR lighting the most repetition of the same team as as as the port and a gain of petition where a man named each was a very forceful person and a British official he'd Kind measurement Maddox refused recalled borders it elevated influence On me that I know of mine but that's too long but what structural steel used much of his work Was This triangle in which you dig a tunnel and take away the middle waters repeatedly she does say he and He's trying to say so because he wrote Italy you find that 1 of the 2 main factors Both Take the churches of a set period of history was precisely this 1 so it died and Mason's Newark but economist picture before he keeps key producer he paid banking that can't do that and you see it because the Fed chief on that at this that these expressed the too fast enough That the form of what he wanted therefore without benefit In this context the he says put his force structure in which used fractals quite spontaneously go back another century to reduce the paint was a very great painter in Japan around 1800 greatest painters of his humanity is well known for his great waiver but look at this picture of their you have Make sure of a very smooth mode Fujian to back these is picture emerges part Of the book for 100 view of Mount Fuji world and on by but if you look at the shapes on the beaches its time then exactly kind of change that people will begin to work toward a track Draw as you go on replaced every 1 of intervals genetic don't you repeated its event stand picture who said this was a different The shadow of a doubt among regions when it was then difficult for painters broke out it's a very curious the only painter who clouds for effectiveness Wasn't there is some among fund eyes Dutch painter in Holland and there is no motives to paint but in clouds so so how can the has the dubious killed in each lending Constable everywhere but the main thing That many painters Mr. butter-flower Please Additional is below the below below below it has I don't know what you would call it a high of structural well is it hoax I had hadn't down pat and you look at the article doors is yours Ellis With some state Substitute line was so was 1 of the simply fires in Over there of thinking that because his case of quarrels involved because he did make made everything you're ready Eva devalue state lines of and the mold for years Or fractal in-depth was loaded Jackson Pollock Last year it was a big pure or because I wonder if it is published an article in Nature of analyzing this picture of ensuring that was
I'm ready ready Most of these problems variances which fractal Shortly afterward somebody phone caches of that thinking which perhaps for about on perhaps not in 1 of the Fed's she decided not products It it was the fact that so different research Notably big book about Top any state that would have had the better he could have gone further because Everything which perfectly being denied the power is the fact that trust the is more severe the and he They had been better quantity and rivets were called to offer the When I'm going to stick to show some pictures of Of this sort but not so fast and make wishes how find track how measure off That's do damage the roughness Of these On the surface actually fought back with these measures have very interesting story on 18 17 do German politicians of Egypt and The independent introduced A way of expressing the roughness of certain functions these functions and naughty added In fact they supposed to be completely of the toward Cuba inventions of human mind but to measure the usual saying that danger or whether they Did not work and Introduced Exporters Italian modified And The simplified generalized is in many ways it Very powerful and then How's introduced notion dimension which he thought was just a matter of matter the Final Four for describing the very fine points of No matter the But turns out that 1 or the other of Of these expressions that it often is the proper way of measuring Graf And that linked to each other And then there's something order from As you begin to New Wave field freely and access to that nature Very complicated that rocks Very subtle better office reviews often publications you find that It is necessary to produce a concept which Sounds that just science fiction with that of negative dimension when places the notion of a name set by notional Lake said which can have different dimensions of course have no time to say more but the topic now Public interest not we can't do a topic which It is is very much at the order of the day At this meeting since the Indian Premier received the medal 19 He was preparing a book that The president mentioned the fractal geometry of nature of it I had a very strong that number and look at brought motion for a very long time around 1900 the 20 people did draw since the pieces of emotions food and then Every so often use picture but by quite clear that the pictures were there purely for and not understand by then I was a killed observer of Complicate changed and so I had my assistant brought very big promotions and to make it a more beautiful and I had ended it started it is not that it is your a white sold Because Give up promotions Years ago it was and the different for funds point of the motion on different blacks homogenate therefore that France is still something of an issue to 4 You're seeing them as you try not Look at everything at the time so at that Where it was your eyes and not much to Europe but then it occurred to me for That's what we find it occurred to me That 1 could simplify But Or lines inside so this schedule is 1 in which the paying blacks but All the tragic stories draw from very far away until they hit Ireland the widening so the Black is a sign that the wine all the points accessible from fitted when it's so this picture based in Italy and member of islands of which we have Generated in our work on geomorphology I'm always very familiar with their South amended that I would give ancient very after of this servers the mission is less than 1 . 5 as Leicester More than 1 . 7 because dimensions his tool measurement of roughness The 2 winners and that farmers part 10 In our eyes if temperature for example had been completely on related to the point of view of hot It would not have been a number but temperature confirms Woodward folks know about hot and he'll Emissions from firms would be skewed by look at that Our things The reaction dimension is a form of the don't because reject it was reported Of which has mentioned it figured they'd rather here a better job From Clancy almost pulled this sense he would completely but by physical argument and then in order Obama approved a correctly And that This year dismantled the youngest in the group sending a very difficult problem 1st people thought I was just the way something for weekend for tomorrow For and for a week a month year essentially trying to 18 years of the The cooperative effort rendering friend he is modifying object Your money your and support It is all where Let me go into your end a rich in his book The a device in fact that's only so many young people came for me shortly after the book came out was But astonishing because nobody expected anything new from this reference number but traffic there from a large number of the book He's tuition of earned yet it's completely fake it on top of
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I was the I was having new a form of time then And I had a very assistant Very good machinery and would make a nice picture depend as it picture I showed you were not a problem on their very hostile conditions because the system was better than a perfect now this was a complicated the religious pictures of And made sure of that Now I understood exactly where they are but when I saw the 1st 79 goes much the anti-missile since what I was doing you trying to do was complete madness Imagine the following year Stupidity you want to study organic chemist or chemist in general but still starting with simple wanted soup which is The the hydrogen water or something you you use start with when not vertical not being a bad 5 If article or or some sort is due Do you get there After you learn about water for example and so basis These pictures I I was trying too much and it is entered the did something core let this chaos rich Thanks All of the Frenchman know about the My was that it A number of its My uncle at younger ages and and I had Tracyton top of this is that was said but they didn't know violent Complicated by simplifying that I so that the testing of their nations where they proceeded but is having fun Everybody knows by now that half of that you get to the final Topic I would discussed and it's a matter of evaluation of a financial crisis it is not a recent interest of mining in fact is a very old interest of mine I have a nite for the 1st time that do work which it created a sensation initiated the wars in the early sixties When finalized financial prices of various kinds but they don't do it and each I was simply a fighting the excessive odds At that time from unity of economists and finance series had Suddenly remember realized that in 1900 very interesting count after Paris named movie he had written Please be patient for a series of speculation and in that The dissertation he produced a process which turned out to be identical to the promotion and coming before end he Postulated that prices for promotion What is Prices followed promotion word very simple you would say that Prices for abortion extraordinary cast of Establishing prices it would have been very much simplified But prices do not follow and example chose this fact strongly and I would like to tell you a little bit about My way in the way of some people of navigating between venture West Maddox and very very Much has In the task to deny I it or lying The line From top From bottom didn't presented being there Actual prices or The imaginative prices simulations of prices according to a model which I've been produced Now Most people Tell the difference because of a boast prices and Imaginative prices of this category special kind of prices have to property that The vanity Every so often a big price changes might be financial prices not prices paid in a grocery store But he drug entirely different and much more Complicates sorry They did not Even the right price of oil which is very much influenced determined by captain Don't pick yet prices of commodities which are just simplified normalized some kind of for example securities stocks bonds or commodities for example American Upland Cotton 56 nature which acquired the Procter & That But after mutated every so often Ford would be a form of devotees these places prices change is by Enormous amounts not once you that much safer and 1 follows a little bit not very closely the markets for example Pharmaceutical companies In the U.S. at this point are subjected right or wrong through very very close supervision of everybody and a education is very good for millions of people but Every 1 Of road 10 out of millions died whether it's so that they would not saw some time ago but role because many had pain reliever which unfortunately the number of people Wendy's became public deprives On the cover of from don't want Not rapidly but it was this Function when some companies that is very some some very good you will Boss from a depressed and go up double because when Tyson knew that a new president is to from new were better For a thief If But dies in shopping Proper accident the Stockman wrote down because of mistakes and don't this sometimes goes up and down for no reason but whatever the reason or that reason 1 must understand that if another study is being placed and fundamentally continues and that is that you it is event it The most Directed tangible aspect of it it not a Fallujah or with a friend of mine who has been for many years But after studying in New York and we ought not for in which were comparable ending this southern for ways of what they would have been if The trend Largest down prices price changes in last year's had not existed so the last 5 years the prices went down many times and you just eliminate the 10 worst at every point in time when The On the condition that more than doubled in other words 10 events as important and as everything it now I would have to stop but it be the discussion of the of prices for something too much More fundamental which is a getting used The notion of a set of measures euros a day jail set of measures 0 when you know if you don't you may not know it but it's a very important quick notions which came up I won't 1900 that details were from the back and and the The principal That the physics use if you have something which is and set with us for a set of measures 0 you fending leg except as it is still because a set of measures here that once the room when after years of you make your own that's not important in these Charm and Nevada I don't think this But is effective in sets of millions sets of measures In does more hours for 1 if that had been motivated by pure mathematics It would have been viewer As very interesting new line caress products But was actually motivated by a very practical A use in the understanding of The number of and Say the depression arises the even when the market is and how we want what happens when on because the usual argument for sectors such as state where the on look more on Four-pronged seam opened on the time booked on lower on everything averages art and nothing is left a gained debate is a very fundamental theorems of over large numbers of the central theory and But the surge immensely important results were A serious told the Phoenician cool conditions are not satisfied this year Carol results so how would you make experiment here to see whether a There Average Word When it's over Jesus Perhaps My profile here is a look at 100 years Where 100 years it's got to be different But couldn't find But It has been equally active that day's prices do not average out there is simply not to finish this Look at this Dagenham said the Dubliner is 1st ones to the original Bashir you process the Burnham motion which is stored interesting mathematical constant tried to use all the time but Which burst Not that because On the top line that just a little bit some points for grows faster than elsewhere but basically it just 4 Within the state and the bottom that many people points off the cluster of her When Let me went to your and assure you think very quickly it's important in orders cases When that begins in total unknown because It was 20 A glut Attention of everybody was somewhat concentrated on promotion that this was the 1st so it would be very important to make capped with simplified picture total but which some elements of the as such you just however Price which interpolate because Whether or not You feel you see on top You go into something which almost as saying the promotion Monday in but something which is worse That In For more stock lost his touch but by studio a single parameter In this case 1 number of nations go from 1 to the other way Here the various Ways of doing it and I 1 of your cue from that here it is here and have a strong dependence of positive or negative all kinds of things just A body A bit And she Something which is perhaps For specialist for people who have a more pointed into this initiative which is when there is a saying that is a quarter of former of the former president The Ronald Reagan of the U.S. who stayed after visiting a forest of Edwards of the Jets 1 national all oldest saying that to you Pieces of brown motion or white noise is very difficult but difference but is 8 pieces of power Exactly same in terms of structure Did you completely different if you make your own analysis you find that different thing right now is to find out same and on The key here is that fact They are not Just a way of taking an old saying is that there are certainly very powerful context where 3 face Our mobile and that there is a friend data But the didn't The main point here This 1 and returned to beginning and that exhibit that often is different here that signs No no and no But if face ain't is not something which new weight in On knowledge of optics Ronald which but makes another and cessation basic plan as they see that but perhaps Accessible to investigations and so Hugh have very simple idea that roughness can be tamed and At this time sure Dubai few pictures of opera and as you can see by Picture factors which are For over the a synthetic this wonderful exhibited the same kind of thinking is Of such extraordinary you'd so It is some hope this instead of a unity of the The order of precise EU made of knowing and Of feeding that Doctor You for 4 baby word it is indispensable to want he gave details of everyday life Are completely different from edition his or People he must do it according few but they behind these differences there are some very basic That is structure No Take A very complicated and dangerous A faster in my life I did not become a member of a new and different from unity i.e. Age actor promise The robots It kind of a living organism which would have vanished role who believed in signs the signs for those 1 said which means that in every few that touched that I had to stop the soon because of strong came incompetent have my contribution in 1st 1st few steps And those The fact that the seat of my life had been so completely different in 1 only a Sokol PGA Nantucket that is what I find that right No this is 7 times that are being invited to Presented Be reduced Russian Congress For the 1st time was in 19 this A key city immediately after the fact the adjournment of nature which the cover on it precisely the Spanish And that and other members Russian Japanese users at that's it I no By that time order was completed The known and respected duly saying the big message which I had been of the mass markets Which has learned in school as being the monster was invented by piano by But kind tour but he's by They may know 1900 D Masters in fact aware of No that all they were just descriptions of the structure of the messiness of nature and of culture it was Not was messages but Fast Sun went on as more people to talk It has proved to be very strong noted activities some books of These books of Hindus who are compromise papers 1 of them amended said 1 of finest 2 others are noises it this book which I published When recently When the initial published a You have a role in Spanish not And the in which I was joined by Richard Parsons who was then a prominent Financial newspaper man who was entirely from that job wrote the book to revel in which My view is already age of prices that and also my use of signs of a N also My use of the addition of mathematics And of a reality you may have noticed Dad On unless made a very bad mistake I did not use the foreign words I never spoke that but I don't you my seven-year plan petition a matter of fact almost nothing that done is classified but must not I certainly have suddenly made mathematical structural and use your examples of many physical structure which acquired the Dudley different matter To be muscle and menu structure was it In living beings vegetal In the and the the stock market I would have run on but Never They don't go by the it's not a matter of judgment it's a matter All of a sudden since perhaps example I think that Petitions with plane It did not A sufficient amount of openness and interest not only in the development of mathematics But In 2 problems can be powerful for example petitioners who comes The stock market and just I see professor of finance what's interesting you would get announced or completed different from the person most potatoes and acetate very interesting these people are very different people some love my work Others don't like know finally understand that after my dogs There be some some sessions of track music and no laughter in the debt A 4th 20 1 of them A somewhat personal a friend of mine Georgia died recently DDT wasn't my generation I think the greatest composer classical more he had the 1st not many periods in which she was showing off of and items music That was mitigated taking too many people but you taught him said that this was just a necessity because music if you don't do something not nobody knows resisted but wants to begin not stuff he went back The basics of music And when he died he had not finished 4th a a series of posters for piano which was hoping would be 24 hour and published he wrote on the adapt Amazing and when they came to me and say that He very much for that I knew very aware of receiving music That he never have understood his music Until he saw my pictures and heard the talks and halt I know nothing about using except that I'm very fond of music familiar music often wins Jordan Don't be That's what she was educated in Romania where she was living in Romania According to What was In effect that the various force throughout YRO 1900 so of the basic things about orchestra instruments He got very sorry but nobody would get him in any form or fashion what distinguishes a piece of music from Noise noise noise he by trial and error better often to bring teacher who was a very forceful and begin man she broke He a piece composition teacher that That's two-way notes Not sense that the emperor boss Told was too many notes But Ms. sense that was fast It is all the time and The Over when they made sure music So he went the other way and he wrote abusive and teachers thought nothing of the did you did go up and down up and down that's enough and then she understood the crafted by belonging to his practice That you must have bad that in every piece of music bad But received no longer so slowly moving seems or or aura of a thick and fast moving down 2 Almost The size of a solemn becomes music dishes was over this told is exactly the music was after that there must be the same kind of structure Did you gave us all world that these days In overall belong listing fees that are separate from different worlds put together in the 1 thing that is safe and that will have understood by musicians in every country I gained from time memorial additional song is from the from Advanced civilizations completely civilizations and the recent songs from a very isolated Funchal as they all have the structure and that structure is Something which you had father or by said without the benefit of any mathematical Okla Course sexual thing but it makes a big difference When such a structure discovered a game I did not discover and friends of mine discovered DDT warning Another day for and alive and other friends were just doing experiments For fund but In all cases The varying Complicated Macy environment in which If you can't have everything they've become an act icon another scientist but mathematics properly understood fans has reduced That is a very small number of very By year's end ID of skating variants and the corresponding tools of which become more and more numerous Our but did important work I wish you would visit His exhibited and thank you for inviting me to breathe and my ideas in this audience