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Les Objets Fractals: Forme, Hasard Et Dimension
Combien mesure la côte de la Bretagne ? Réfléchissez, et vous vous méfierez des dictionnaires. Quelle est la forme de la montagne ou du nuage que voici ? La simplicité de ces questions est trompeuse. Que peut-on dire pour caractériser les formes créées par le chaos ? C'est l'auteur de ce livre qui, le premier, a trouvé le moyen de soumettre ces questions à la démarche scientifique, et il a montré qu'il y a entre elles une affinité profonde et surprenante. Prenant comme base certains objets dont la forme est très rugueuse, très poreuse ou très fragmentée, objets qu'il a appelés fractales, Benoît Mandelbrot a conçu, développé et utilisé une nouvelle géométrie de la nature et du chaos. Elle a désormais pris une très grande extension, apprenant au savant et à l'ingénieur - et à d'autres ! - à voir le monde de façon nouvelle. Le langage fractal a également un impact, aussi puissant qu'imprévu, sur l'art populaire et les mathématiques pures. Ce livre fut le premier exposé de la géométrie fractale ; il reste un document historique autant qu'une introduction de choix.
216 pages, Paperback
First published January 1, 1975
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724 people want to read
About the author
Benoît B. Mandelbrot
27 books316 followersBenoît B. Mandelbrot, O.L.H., Ph.D. (Mathematical Sciences, University of Paris, 1952; M.S., Aeronautics, California Institute of Technology, 1949) was a mathematician best known as the father of fractal geometry. He was Sterling Professor Emeritus of Mathematical Sciences at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Center; and Battelle Fellow at the Pacific Northwest National Laboratory.
Mandelbrot was born in Poland, but his family moved to France when he was a child; he was a dual French and American citizen and was educated in France. He has been awarded with numerous honors, including induction into the Legion d'honneur, as well as the 1986 Franklin Medal for Physics, the 1993 Wolf Prize for Physics, the 2000 Lewis Fry Richardson Medal of the European Geophysical Society, and the 2003 Japan Prize "for the creation of universal concepts in complex systems."
Mandelbrot was born in Poland, but his family moved to France when he was a child; he was a dual French and American citizen and was educated in France. He has been awarded with numerous honors, including induction into the Legion d'honneur, as well as the 1986 Franklin Medal for Physics, the 1993 Wolf Prize for Physics, the 2000 Lewis Fry Richardson Medal of the European Geophysical Society, and the 2003 Japan Prize "for the creation of universal concepts in complex systems."
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Displaying 1 - 9 of 9 reviews
November 25, 2023
If you try to measure the length of a river or the coast line of a country, you’ll run into a technical challenge. The length is not a fixed value. Rather, it depends on the ruler you use. The more fine-grained the measurement unit, the longer the river or coast line because you now find yourself following ever more fine-grained zigzagging. Pushed to extreme, with an arbitrarily tiny ruler, you’ll find the river length to be infinite. This is the puzzle that led to the discovery and more systematic understanding of what is now called fractals.
If we back off a little bit, a 1-dimensional object such as a line segment has length. But a 2-dimensional object (area covered by a square) does not have a length. Indeed, you can fit an infinitely long squiggly curve inside a square of finite area. We knew this from school. But in school, we deal with shapes of nice properties that Mandelbrot calls Euclids (squares, balls, etc). In real life, objects have weird properties. Take a simple example of a Koch snowflake: start with an equilateral triangle; take every one of the sides and divide it into three thirds; using a middle third as a base for a smaller equilateral triangle pointing towards the outside of the big triangle and then erase the base of this smaller triangle (the middle third of the side). After one iteration of this action, you get a 12-sided “sharp snowflake”. Now do the same to every one of the 12 sides, you get a better snow flake. You can keep doing this ad infinitum. Every time you do so, the total “coast line” increases by a factor of 4/3. So infinite iterations later you get a Koch snow flake. The perimeter/coast line of the object is infinite and yet the object has a finite area, in stark contrast to a square, a circle or any other nice Euclid. A closer inspection shows that the coast line does not even have integer dimension. The Koch snowflake for instance has dimensionality of 1.26. Mandelbrot coined the word “fractal” to describe these objects with fractional dimensions.
It turns out, nature is full of such objects. Coast lines, for instance, have an empirically measured dimension of about 1.2.
The concept of fractals is fascinating. This book, however, is quite technical and is probably meant for those who have a very deep interest in the subject.
If we back off a little bit, a 1-dimensional object such as a line segment has length. But a 2-dimensional object (area covered by a square) does not have a length. Indeed, you can fit an infinitely long squiggly curve inside a square of finite area. We knew this from school. But in school, we deal with shapes of nice properties that Mandelbrot calls Euclids (squares, balls, etc). In real life, objects have weird properties. Take a simple example of a Koch snowflake: start with an equilateral triangle; take every one of the sides and divide it into three thirds; using a middle third as a base for a smaller equilateral triangle pointing towards the outside of the big triangle and then erase the base of this smaller triangle (the middle third of the side). After one iteration of this action, you get a 12-sided “sharp snowflake”. Now do the same to every one of the 12 sides, you get a better snow flake. You can keep doing this ad infinitum. Every time you do so, the total “coast line” increases by a factor of 4/3. So infinite iterations later you get a Koch snow flake. The perimeter/coast line of the object is infinite and yet the object has a finite area, in stark contrast to a square, a circle or any other nice Euclid. A closer inspection shows that the coast line does not even have integer dimension. The Koch snowflake for instance has dimensionality of 1.26. Mandelbrot coined the word “fractal” to describe these objects with fractional dimensions.
It turns out, nature is full of such objects. Coast lines, for instance, have an empirically measured dimension of about 1.2.
The concept of fractals is fascinating. This book, however, is quite technical and is probably meant for those who have a very deep interest in the subject.
December 27, 2015
Great book !
This entire review has been hidden because of spoilers.
August 10, 2021
Intéressant, sans plus.. la faute à une description du livre qui le vend comme « une porte d’entrée dans le monde des objets fractals », un livre pour novices donc, à priori. Pourtant, je trouve le livre assez indigeste, et ce que j’y ai appri, j’aurais pu l’apprendre en 45 min de lecture à peu près.
Afin que mon commentaire soit plus parlant, il faut savoir que je n’ai pas un niveau zéro en maths. J’ai un Master en physique appliquée et ingénierie physique, donc des maths j’en ai pas mal fait, mais ici le problème c’est que les formules sont données sans vraiment d’esquisse de démonstration, et même si l’auteur semble parfaitement assumer ce choix, je le trouve personnellement dommageable. On a l’impression de lire un livre en développement, non finalisé.
Pour ceux vraiment intéressés par le sujet, je conseillerai plutôt « La théorie du Chaos » de James Gleick.
Afin que mon commentaire soit plus parlant, il faut savoir que je n’ai pas un niveau zéro en maths. J’ai un Master en physique appliquée et ingénierie physique, donc des maths j’en ai pas mal fait, mais ici le problème c’est que les formules sont données sans vraiment d’esquisse de démonstration, et même si l’auteur semble parfaitement assumer ce choix, je le trouve personnellement dommageable. On a l’impression de lire un livre en développement, non finalisé.
Pour ceux vraiment intéressés par le sujet, je conseillerai plutôt « La théorie du Chaos » de James Gleick.
December 29, 2021
Souvent trop difficile pour moi.
J'en ai retenu les considérations sur la mesure des côtes et les différentes solutions pratiques que l'on pouvait y apporter, le calcul de la dimension d'homothétie, ce qui m'a permis de comprendre (enfin) comment une dimension pouvait ne pas être entière, et, de façon sous-jacente, comment définir une dimension.
A part ça, bien des chapitres commencent en exposant un problème intéressant et concret dont ensuite je ne comprends pas le traitement.
Un des chapitres qui m'a rassérénée à la fin est celui, très mathématique, où l'auteur reprend les différentes définitions ( et modes de calcul), de la dimension.
Dans l'ensemble j'ai trouvé ce livre difficile mais instructif. Je pense que j'aurai du plaisir à y revenir, pour progresser un tant soit peu dans mon assimilation des concepts exposés.
J'en ai retenu les considérations sur la mesure des côtes et les différentes solutions pratiques que l'on pouvait y apporter, le calcul de la dimension d'homothétie, ce qui m'a permis de comprendre (enfin) comment une dimension pouvait ne pas être entière, et, de façon sous-jacente, comment définir une dimension.
A part ça, bien des chapitres commencent en exposant un problème intéressant et concret dont ensuite je ne comprends pas le traitement.
Un des chapitres qui m'a rassérénée à la fin est celui, très mathématique, où l'auteur reprend les différentes définitions ( et modes de calcul), de la dimension.
Dans l'ensemble j'ai trouvé ce livre difficile mais instructif. Je pense que j'aurai du plaisir à y revenir, pour progresser un tant soit peu dans mon assimilation des concepts exposés.
February 3, 2021
Ho trovato il libro interessante ma troppo difficile per un non specialista. L’argomento è certamente complesso e semplificarlo senza banalizzarlo è un’impresa difficile, ma Mandelbrot, complice una prosa piuttosto barocca, non fa grossi sforzi in questo senso. I concetti matematici necessari a comprendere il libro non sono esposti chiaramente e anche quelli nuovi che l’autore introduce sono spesso esposti in maniera oscura o dati per acquisiti. Probabilmente un matematico lo troverebbe bello interessante ma io che non lo sono, anche se non sono nemmeno del tutto a digiuno della materia, ho fatto grossi sforzi per comprenderlo, spesso senza riuscirci. Non è di certo una lettura introduttiva sui frattali, richiede un grosso bagaglio di conoscenze.
October 7, 2024
Sugestão do meu pai. Livro que me faz sentir burro, mas é para mostrar aos outros que sou inteligente. As partes mais interessantes para mim foram todos os aspetos da Natureza que podem ser explicados pelos fractais, apesar do autor entrar mais em detalhe da teoria por trás do que explicar o quão bem é que os modelos se adequam ou porquê. Além disso tem figuras giras e faz-te pensar de uma forma diferente. É sempre engraçado ver como é que funciona a investigação e o desenvolvimento de outras áreas da ciência, como a matemática.
September 10, 2024
C'est toujours un peu vexant quand le livre s'annonce "grand public" mais qu'au bout d'un tiers on ne comprend rien... Je n'ai pas pu finir le livre à cause de ça, mais j'ai remarqué que la bibliographie contenait environ 5 pages de références à ... lui-même ! La science des fractales est superbe mais c'est pas avec ce livre que j'en ai vraiment profité.
May 25, 2024
Livre intéressant, l'auteur raccroche les concepts à des phénomènes physiques. Mais la lecture n'est définitivement pas aisée
March 7, 2025
Listen, the ideas are super interesting. It’s striking the number of varied ideas he explains using the (then) new idea of fractals. It’s so boring though, hahaha. It’s just too close to being a math textbook. Maybe if that’s what I had planned I would have been better prepared, but if you want to just read it it’s going to be a grind. Still rated highly because the ideas are so interesting, and I do think about them still.
Displaying 1 - 9 of 9 reviews