The discovery of fractals
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The discovery of fractals |
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for this page
The history of fractals begins with Benoît Mandelbrot. Nevertheless, as far as a certain number of things were known before his works, these have to be pointed out so that we can better see what his contribution was.
This is no more than telling a story, in an incomplete way, at second... or third hand, and intended to provide a few landmarks for those who just discover this domain. In some cases I do not have the precise dates of the work mentioned and therefore, I would be grateful to anyone who could help me to improve this paper.
What follows is not the history of fractals but, in a more limited way, the history of their discovery.
What was known before Mandelbrot |
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| The knowledge can be grouped into three fields : natural objects, geometrical figures, and mathematical theories. Of course this distinction is quite artificial, especially for the last two groups, but it will simplify the account. | |
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The first fractals described are dated at the end of the XIXth century. |
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In 1890 Peano published his famous curve and in 1891, Hilbert published a quite similar curve but, may be, less well-known. |
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| Peano's curve after 3 iterations | Hilbert's curve after 4 iterations |
Koch's curve was published in 1904 and the title of the paper is well worth to be mentioned : "On a continuous curve without any tangent, obtained through an elementary geometrical construction" (according to the Mandelbrot's book bibliography, the original title is in French). |
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| Sierpinski's gasket is more recent (1915 : thanks to William Mcworter for the date). |
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Natural objects |
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| To illustrate the fractal nature of some objects or natural phenomena two examples are enough. The first one is a remarkable text drawn from the foreword of Jean Perrin's Les atomes (1913). This text being fairly long, I have reproduced its translation on a separate page. Benoît Mandelbrot mentions it in the introduction of his book Les objets fractals and says that this text has had a considerable influence on the works of Norbert Wiener on
Brownian movement. Mandelbrot also says that Wiener's works have been his main source of inspiration but that he only discovered personally Perrin's text after he had started writing his own book. Let us remember from this that some colloids have
a fractal structure and that the trajectories of particles submitted to Brownian
movements are also fractal.
The second example is particularly interesting for several reasons : at first, it is more concrete ; second, it is evoked with much sagacity by Perrin, but it only seems to have been studied concretely by Richardson (1961), who probably didn't know Perrin's text ; finally, it played an important part in Mandelbrot's work. Let us set the problem the way Mandelbrot did it : How long is the coast of Britain ? The question is less trivial than it seems. Indeed, a dented coast has gulfs and promontories. Each gulf itself is made of smaller bays and each promontory of smaller ones. One can guess that the length found will depend on the degree of detail one has decided to take into account. The inescapable consequence is that the length of the coast of Britain is infinite, a result apparently paradoxical, but that had clearly be stated by Perrin without any mathematical justification.
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Mathematics |
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| What was then known in the purely mathematical field ? The existence of functions without derivatives was known but this was not well considered by some mathematicians who thought this was almost an aberration. On the other hand, the problem of the dimension of geometrical figures had been the object of interesting works. We know that a point as a dimension 0 ; a line has a dimension 1 ; a surface, 2 ; a volume, 3. These are said to be
Euclidian or topological dimensions.
But a few mathematicians noticed that there were more sophisticated means to define the dimension of an object. Fundamental work was done by Hausdorff (1919), then developed by Besicovitch (1935). The Hausdorff-Besicovitch dimension has played , later on, a major role in the domain of fractals. The last mathematics field I would like to evoke is the one of the iteration of complex polynomials, independently studied by Julia (1918) and Fatou (1919-1920). In short one can say that Julia sets are the boundaries of Fatou domains. Except for particular cases, Julia sets are fractals and we can marvel at the fact that Julia and Fatou were able to determine various properties of these sets without a graphic computer. All the works on turbulence and chaos should also be mentioned (particularly Ruelle) because they contain many fractal elements. But this remark could also have been made in the paragraph about natural objects or phenomena.
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Mandelbrot's contribution |
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The arising of the fractal concept |
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| Everything mentioned here above was well known before Mandelbrot's works, but these were scattered elements, sometimes corresponding to cases considered as mathematical curiosities, or known only by a small number of specialists (this was the case for non topological dimensions). Therefore little attention had been paid to this knowledge an no one had thought of bringing together all theses elements. Mandelbrot's merit was to bring them together and to develop an entirely new mathematical domain. The term self-similar seems to have appeared for the first time in 1964, in an internal report at IBM (where Mandelbrot was doing research) and in the title of a 1965 paper. But the certificate of baptism of the word fractal is dated 1975, since it was created for the first edition of the book Les objets fractals. Nevertheless it is not this book that gives the best chronological view on the discovery of the fractal concept. When one reads the list of Mandelbrot's publications between 1951 and 1975, date when the French version of his book was published, one is astonished by the variety of the studied fields : noise on telephone lines, games theory, linguistics, economy, cosmology, turbulence... The multiplicity of his fields of interest has undoubtedly played a key role in the genesis of his discovery. In one of his interviews (How I discovered fractals - La Recherche, 1986) Mandelbrot tells us that in 1962 he was interested in the mathematical problem related to the distribution of incomes, to the fluctuation of prices and to the stock exchange rates. The classical theories estimated that short term fluctuations were due to speculation (and were largely random) whereas long term fluctuations reflected the fundamental laws of economy. And yet, like others, Mandelbrot noticed that the distribution of the random fluctuations did not obey to a normal law. He emitted the hypothesis that there was no statistical differences between short term and long term fluctuations. On this basis he was able to develop a mathematical model that could stimulate purely fictive, but indeed very realistic, stock exchange chronicles. Mandelbrot was then interested in noise on telecommunications lines, in turbulence, in geophysical problems such as the length of coastlines, in the hydrologic regime of streams that were badly described with the theories known at that time... in all these cases he was capable of applying the same mathematical approach and he found every time the notion of self-similarity.
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Fractals, concrete objects or phenomena, probability |
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| The case of transmissions errors on the connecting lines of computers enables us to understand well Mandelbrot's approach. All telecommunications lines are subjected to random fluctuations that make the background noise. The latter disturbs more or less the transmission of messages. The case of computer lines is interesting because the information circulates there in the way of bit having a fixed magnitude. If the background noise remains below a certain value it does not disturb the message. If it fluctuates above this threshold, some bits could be altered and change from 0 to 1, or from 1 to 0. When examining the errors one can notice that they happen in batches separated by quiet intervals of variable length. When examining one batch
of errors, one notice that it is made of smaller batches that are also separated by quiet periods without error. In other words we have here a random phenomenon with some kind of internal similarity . With a little imagination, this linear phenomenon made of points (the faulty bits) reminds us of Cantor dust, the elements of witch would have been mixed up. Mandelbrot studied the mathematical process that enables us to create random Cantor dust describing perfectly well the fractal structure of the batches of errors on computer lines.
This example enables us to outline two important characteristics of Mandelbrot's works.
The second characteristic is that almost all models used by Mandelbrot are of probabilistic nature and therefore are an extension of the probabilities theories. Mandelbrot did not start applying the concept of fractal to deterministic domains before 1979-1980. This gives me the opportunity to go back to Mandelbrot's important contribution to the problem of the length of coasts in his paper How long is the coast of Britain. Statistical self-similarity and fractional dimension (Science, 155, 636-638 ; 1968). The author starts from the results of Richardson's paper, that was quite unknown. Where the latter only saw in his formula an empirical exponent
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Who discovered the Mandelbrot set ? |
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What follows has been inspired, although not totally, by that Mandelbrot tells in the third edition of is book Les objets fractals.
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| People seem to agree to credit Brooks and Matelski (1981) for the first image of M, but a few details deserve to be pointed out. At first, the 1981 date (1980 according to other sources ; I did not check the original paper), which enables Mandelbrot to say that this work is roughly contemporary to his, corresponds to the date of publication of the proceedings of a conference that took place in 1978, therefore before Mandelbrot's work. This one affirms that the fuzzy picture of M published on this occasion was obtained by a computer scientist, friend of both authors and that the latter did not pay much attention to it, unlike the approach
Mandelbrot says he has used. It is obvious that the notion of anteriority is not very clear in this story. But is it really important ? The first published image was not from Mandelbrot, even if he is the one that has most used the graphic resources of computers for is study. What is certain is that this set was called Mandelbrot set by Douady and Hubbard in 1982 because they didn't know Brooks and Matelski's paper. Finally, it was them who wrote it for the first time in its canonical form (iteration of the complex polynomial  ). What is also certain is that the work done on M was only a minute part, even after 1980, of Mandelbrot's research on fractals.
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What definition can be given for fractals ? |
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| Mandelbrot insists in his book on the fact that he will only give an empirical definition of fractals, no abstract definition being totally satisfactory. For instance it is said that fractals are objects with fractional dimension (Mandelbrot himself has used this definition at a certain time and he says, in his interview published in La Recherche, that the term fractal was chosen to evoke fractional). But he agrees that this is wrong twice. At first this dimension can be an irrational number ; and moreover it can be an integer. For example the dimension of the M boundary is 2, just like the
Brownian trajectories (with nevertheless an important
difference : the M set is contained in a plane whereas
Brownian trajectories develop in a 3 dimensional space). A definition that would not be so bad would be to say that fractals are objects whose Hausdorff-Besicovitch dimension (or other) is greater than their topological
(Euclidian) dimension but, according to Mandelbrot, this seems to exclude a few objects that are real fractals.
In the end, all that remains is the notion of self-similarity. |
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(in Les objets fractals Mandelbrot write
but this is not coherent with what follows in his text) where L is the length of coast approximated for the step
, the exponent
being variable from one coast to another, but always much inferior to 1. Under these conditions the function tends towards the infinite when the step tends towards 0.