Benoit Mandelbrot, el matemático que amplió el concepto de geometríaBenoit Mandelbrot, matemático polaco, falleció, como comentábamos el otro

día, el pasado 14 de octubre, aunque hasta el día 16 no nos enteramos de esta

triste noticia. Mandelbrot es, como a mí me gusta decir, el último grande, una

de las pocas personas que ha sido capaces de crear una nueva rama de las

matemáticas, la geometría fractal, con gran interés tanto por la teoría como

por las aplicaciones de los resultados obtenidos.

Notas biográficasBenoit Mandelbrot nació en Varsovia el 20 de noviembre de 1924 dentro de una

familia con cierta tradición académica (aunque su padre se ganaba la vida con la

compra-venta de ropa). Fueron dos tíos suyos quienes se encargaron de

introducir a Mandelbrot en el mundo de las matemáticas. Uno de ellos, Szolem

Mandelbrojt, se encargó de su educación cuando la familia Mandelbrot emigró a

Francia en 1936.

El hecho de que Mandelbrot estudiara en la época de la Primera Guerra Mundial,

entre otras cosas, provocó que su educación no fuera convencional. El propio

Mandelbrot atribuye gran parte de su éxito matemático a esta educación poco

convencional, ya que ello le permitió pensar de forma distinta a la que se le suele

inculcar a quien sigue la educación habitual. Su gran visión e intuición geométrica

también contribuyeron a ello.

Después de estudiar en Lyon y permanecer un día en la École Normale de París,

Mandelbrot comenzó sus estudios en la École Polytechnique en 1944 bajo la

dirección dePaul Lévy, quien también ejerció gran influencia en él. Más adelante

se doctoró en la Universidad de París y viajó a Estados Unidos, donde, entre otras

cosas, fue el último estudiante de postdoctorado de John Von Neumann.

Echando un ojo a los mentores de Mandelbrot podemos ver que la lista no tiene

desperdicio, si uno era bueno el siguiente era mejor.

A lo largo de su vida fue profesor en la Universidad de Harvard y en la

Universidad de Yale (donde terminó su carrera), entre otras instituciones. Pero

posiblemente fue su trabajo en IBM en el Centro de Investigaciones Thomas B.

Watson de Nueva York lo que más le ayudó en sus estudios, ya que allí le

brindaron libertad total en sus investigaciones.

¿Cuánto mide la costa de Gran Bretaña?Benoit Mandelbrot es el padre de la denominada Geometría Fractal, una nueva

rama de la geometría que podemos decir que estudia los objetos tal como son.

Mandelbrot pensó que las cosas en la realidad no son tan perfectas como las

muestra la geometría euclídea: las esferas no son realmente esferas, las líneas no

son perfectamente rectas, las superficies no son uniformes… Ello le llevó a

estudiar estas imperfecciones, derivando estos estudios en la creación

de esta nueva rama de la geometría.

Las primeras ideas sobre fractales de Mandelbrot fueron publicadas en la

revista Scienceen 1967 a través de su artículo ¿Cuánto mide la costa de Gran

Bretaña? En él da ciertas evidencias empíricas de que la longitud de una línea

geográfica (como por ejemplo, la costa de Gran Bretaña) depende de la regla

con la que la midamos. En líneas generales, la costa tendrá mayor

longitud cuanto menor sea la unidad de medida utilizada, esto es,

cuanto más cerca estemos mirando a la costa mayor longitud tendrá.

También habla de ciertas curvas autosemejantes, es decir, curvas que son

semejantes a una parte de ellas mismas. Por ejemplo, las propias costas son un

ejemplo de ello (no un ejemplo exacto, pero sí lo suficientemente aproximado

como para comprender de qué estamos hablando), ya que la estructura

quebradiza de las mismas hace que si vemos una porción de costa y

después hacemos zoom en esa zona, lo que vemos en ese momento tiene una

forma semejante a la primera porción que observamos.

El caso es que este tipo de objetos se salían de la concepción euclídea de la

geometría. Es posible que por ello Mandelbrot buscara un nuevo término para

designarlo: fractal (del latín fractus: quebrado, fracturado), que acuñó en 1975.

Aunque ha habido diversos debates sobre cómo definirlo de forma clara y

concisa, podemos decir que un fractal es precisamente eso, un objeto cuya

estructura se repite a diferentes escalas. Y tanto se salen estos

objetos autosemejantes de la geometría euclídea que generalmente tienen

dimensiones no enteras. Por poner un ejemplo, una línea recta tiene dimensión 1

y un plano tiene dimensión 2, pero la costa occidental de Gran Bretaña tiene,

aproximadamente, dimensión 1,25.

Mandelbrot publicó más tarde The Fractal Geometry of Nature, donde amplió y

actualizó sus ideas sobre los fractales. La manera apasionada de escritura y el

gran énfasis en la intuición visual y geométrica que impregnaba a esta

publicación hizo que terminara por popularizarse tanto entre los estudiosos del

tema como entre el público en general. El hecho de que Mandelbrot apoyara sus

ideas con gráficos e ilustraciones también contribuyó a ello.

El conjunto de MandelbrotEs interesante comentar que fue su tío Szolem quien, posiblemente sin querer, le

indujo a introducirse en el mundo fractal mostrándole unos estudios de Gaston

Julia sobre 1945. En su momento a Mandelbrot ni siquiera le gustaron, pero más

adelante se volvió a encontrar con ellos y comenzó sus estudios sobre el

conocido como conjunto de Julia, y también del actualmente

denominado conjunto de Mandelbrot.

Conjunto de Mandelbrot

Este conjunto es, en líneas generales, el conjunto de puntos para los cuales cierta operación matemática da siempre resultados menores que un cierto valor. Más concretamente:

Un número complejo   (un punto del plano, vamos) está en el conjunto de

Mandelbrot si la sucesión de puntos siguiente

está acotada, es decir, si esta sucesión no tiende a infinito (esto es, el valor

de sus términos tiene un “tope” que ninguno de ellos sobrepasa).

Si esta sucesión de puntos no está acotada, o lo que es lo mismo, sus valores

crecen y crecen indefinidamente, el punto no está en el conjunto.

Los puntos del conjunto de Mandelbrot son los que aparecen en negro en la

imagen anterior. Los que no pertenecen al conjunto no tienen por qué

representarse, aunque lo que le da ese tremendo juego a este conjunto es

representar con colores la velocidad con la que la sucesión anterior se acerca a

infinito, como aparece también en la imagen. Los puntos para los que su sucesión

crece muy rápido están representados en color rojo intenso. El rojo va tornándose

más suave conforme la velocidad de crecimiento es menor. Los puntos muy

cercanos al conjunto (en blanco) son puntos para los que ha hecho falta calcular

muchísimos valores de la sucesión asociada para ver que no está acotada. Este

juego de colores provoca que al hacer zoom en el conjunto, las imágenes que se

crean sean de una belleza inusitada (¿fórmulas matemáticas creando obras de

arte? Que raro…¿o no?). Además, este zoom hace que nos demos cuenta de

esaautosemejanza de la que hablábamos hace un rato, ya que al acercarnos

vemos que el propio conjunto contiene copias exactas de si mismo.

Para comprobar estos dos apuntes os recomiendo ver este vídeo. Es algo largo

(10 minutos), pero os aseguro que merece mucho la pena. Mucho cuidado con él,

no os vayáis a marear:

Y para terminar, os quiero enseñar un par de imágenes que ilustran a la

perfección el amor que mucha gente le tiene a esta disciplina (la geometría

fractal) y en particular a este interesantísimo conjunto:

El conjunto de Mandelbrot en…un campo

El conjunto de Mandelbrot en…una espalda

Benoit Mandelbrot

Born: 20 November 1924 in Warsaw, PolandDied: 14 October 2010 in Cambridge, Massachusetts, USA

Click the picture aboveto see six larger pictures

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Benoit Mandelbrot was largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both mathematics and elsewhere in nature.

Mandelbrot was born in Poland in 1924 into a family with a very academic tradition. His father, however, made his living buying and selling clothes while his mother was a doctor. As a young boy, Mandelbrot was introduced to mathematics by his two uncles.

Mandelbrot's family emigrated to France in 1936 and his uncle Szolem Mandelbrojt, who was Professor of Mathematics at the Collège de France and the successor of Hadamard in this post, took responsibility for his education. In fact the influence of Szolem Mandelbrojt was both positive and negative since he was a great admirer of Hardy and Hardy's philosophy of mathematics. This brought a reaction from Mandelbrot against pure mathematics, although as Mandelbrot himself says, he now understands how Hardy's deep felt pacifism made him fear that applied mathematics, in the wrong hands, might be used for evil in time of war.

Mandelbrot attended the Lycée Rolin in Paris up to the start of World War II, when his family moved to Tulle in central France. This was a time of extraordinary difficulty for Mandelbrot who feared for his life on many occasions. In [3] the effect of these years on his education was emphasised:-

The war, the constant threat of poverty and the need to survive kept him away from school and college and despite what he recognises as "marvellous" secondary school teachers he was largely self taught.

Mandelbrot now attributed much of his success to this unconventional education. It allowed him to think in ways that might be hard for someone who, through a

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conventional education, is strongly encouraged to think in standard ways. It also allowed him to develop a highly geometrical approach to mathematics, and his remarkable geometric intuition and vision began to give him unique insights into mathematical problems.

After studying at Lyon, Mandelbrot entered the École Normale in Paris. It was one of the shortest lengths of time that anyone would study there, for he left after just one day. After a very successful performance in the entrance examinations of the École Polytechnique, Mandelbrot began his studies there in 1944. There he studied under the direction of Paul Lévy who was another to strongly influence Mandelbrot.

After completing his studies at the École Polytechnique, Mandelbrot went to the United States where he visited the California Institute of Technology. After a Ph.D. granted by the University of Paris, he went to the Institute for Advanced Study in Princeton where he was sponsored by John von Neumann.

Mandelbrot returned to France in 1955 and worked at the Centre National de la Recherche Scientific. He married Aliette Kagan during this period back in France and Geneva, but he did not stay there too long before returning to the United States. Clark gave the reasons for his unhappiness with the style of mathematics in France at this time [3]:-

Still deeply concerned with the more exotic forms of statistical mechanics and mathematical linguistics and full of non standard creative ideas he found the huge dominance of the French foundational school of Bourbaki not to his scientific tastes and in 1958 he left for the United States permanently and began his long standing and most fruitful collaboration with IBM as an IBM Fellow at their world renowned laboratories in Yorktown Heights in New York State.

IBM presented Mandelbrot with an environment which allowed him to explore a wide variety of different ideas. He has spoken of how this freedom at IBM to choose the directions that he wanted to take in his research presented him with an opportunity which no university post could have given him. After retiring from IBM, he found similar opportunities at Yale University, where he is presently Sterling Professor of Mathematical Sciences.

In 1945 Mandelbrot's uncle had introduced him to Julia's important 1918 paper claiming that it was a masterpiece and a potential source of interesting problems, but Mandelbrot did not like it. Indeed he reacted rather badly against suggestions posed by his uncle since he felt that his whole attitude to mathematics was so different from that of his uncle. Instead Mandelbrot chose his own very different

course which, however, brought him back to Julia's paper in the 1970s after a path through many different sciences which some characterise as highly individualistic or nomadic. In fact the decision by Mandelbrot to make contributions to many different branches of science was a very deliberate one taken at a young age. It is remarkable how he was able to fulfil this ambition with such remarkable success in so many areas.

With the aid of computer graphics, Mandelbrot who then worked at IBM's Watson Research Center, was able to show how Julia's work is a source of some of the most beautiful fractals known today. To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics.

The Mandelbrot set is a connected set of points in the complex plane. Pick a point z0 in the complex plane.

Calculate:z1 = z0

2 + z0

z2 = z12 + z0

z3 = z22 + z0

. . .

If the sequence z0 , z1 , z2 , z3 , ... remains within a distance of 2 of the origin forever, then the point z0 is said to be in the Mandelbrot set. If the sequence diverges from the origin, then the point is not in the set.

You can see the Mandelbrot set

His work was first put elaborated in his book Les objets fractals, forn, hasard et dimension (1975) and more fully in The fractal geometry of nature in 1982.

On 23 June 1999 Mandelbrot received the Honorary Degree of Doctor of Science from the University of St Andrews. At the ceremony Peter Clark gave an address [3] in which he put Mandelbrot's achievements into perspective. We quote from that address:-

... at the close of a century where the notion of human progress intellectual, political and moral is seen perhaps to be at best ambiguous and equivocal there is one area of human activity at least where the idea of, and achievement of, real progress is unambiguous and pellucidly clear. That is mathematics. In 1900 in a famous address to the International Congress of mathematicians in Paris David Hilbert listed some 25 open problems of outstanding significance. Many of

those problems have been definitively solved, or shown to be insoluble, culminating as we all know most recently in the mid-nineties with the discovery of the proof of Fermat's Last Theorem. The first of Hilbert's problems concerned a thicket of issues about the nature of the continuum or the real line, a major concern of 19th and indeed of 20th century analysis. The problem was both one of geometry concerning the nature of the line thought of as built up of points and of arithmetic thought of as the theory of the real numbers. The integration of those two fields was one of the great achievements of Richard Dedekind and Georg Cantor, the latter of whom we [St Andrews University] were intelligent enough to honour in 1911.

Now lurking about so to speak in the undergrowth of that achievement lay certain very extraordinary geometric objects indeed. To all at the time, they seemed strange, indeed rather pathological monsters. Odd indeed they were, there were curves - one dimensional lines in effect - which filled two dimensional spaces, there were curves which were well behaved, that is nice and continuous but which had no slope at any point (not just some points, ANY points) and they went by strange names, the Peano Space filling curve, the Sierpinski gasket, the Koch curve, theCantor Ternary set. Despite their pathological qualities, their extraordinary complexity, especially when viewed in greater and greater detail, they were often very simple to describe in the sense that the rules which generated them were absurdly simple to state. So odd were these objects that mathematicians set about barring these monsters and they were set aside as too strange to be of interest. That is until our honorary graduand created out of them an entirely new science, the theory of fractal geometry: it was his insight and vision which saw in those objects and the many new ones he discovered, some of which now bear his name, not mathematical curiosities, but signposts to a new mathematical universe, a new geometry with as much system and generality as that of Euclid and a new physical science.

As well as IBM Fellow at the Watson Research Center Mandelbrot was Professor of the Practice of Mathematics at Harvard University. He also held appointments as Professor of Engineering at Yale, of Professor of Mathematics at the École Polytechnique, of Professor of Economics at Harvard, and of Professor of Physiology at the Einstein College of Medicine. Mandelbrot's excursions into so many different branches of science was, as we mention above, no accident but a very deliberate decision on his part. It was, however, the fact that fractals were so widely found which in many cases provided the route into other areas [3]:-

I should not ... give the impression that we have here before us a mathematician alone. Let me explain why. The first of his great insights was the discovery that the extraordinarily complex almost pathological structures, which had been long

ignored, exhibited certain universal characteristics requiring a new theory of dimension to treat them adequately which he had generalised from earlier work of Hausdorff and Besicovitch but the second great insight was that the fractal property so discovered, the general theory of which he had provided, was present almost universally in Nature. What he saw was that the overwhelming smoothness paradigm with which mathematical physics had attempted to describe Nature was radically flawed and incomplete. Fractals and pre-fractals once noticed were everywhere. They occur in physics in the description of the extraordinarily complex behaviour of some simple physical systems like the forced pendulum and in the hugely complex behaviour of turbulence and phase transition. They occur as the foundations of what is now known as chaotic systems. They occur in economics with the behaviour of prices and as Poincaré had suspected but never proved in the behaviour of the Bourse or our own Stock exchange in London. They occur in physiology in the growth of mammalian cells. Believe it or not ... they occur in gardens. Note closely and you will see a difference between the flower heads of broccoli and cauliflower, a difference which can be exactly characterised in fractal theory.

Mandelbrot has received numerous honours and prizes in recognition of his remarkable achievements. For example, in 1985 Mandelbrot was awarded the Barnard Medal for Meritorious Service to Science. The following year he received the Franklin Medal. In 1987 he was honoured with the Alexander von Humboldt Prize, receiving the Steinmetz Medal in 1988 and many more awards including the Légion d'Honneur in 1989, the Nevada Medal in 1991, the Wolf prize for physics in 1993 and the 2003 Japan Prize for Science and Technology.

A full list of his prizes and honours is available (as a download) at this link.

Article by: J J O'Connor and E F Robertson

Click on this link to see a list of the Glossary entries for this page

La Geometría Fractal de la Naturaleza (The Fractal Geometry of Nature) - MANDELBROT

Hola Taringueros, hoy he encontrado este libro de Benoit Mandelbrot, sobre la aplicación de los fractales en la naturaleza y como las funciones matemáticas pueden generar aplicaciones vivas que encontramos todos los días alrededor del mundo. Desafortunadamente para algunos EL LIBRO SE ENCUENTRA EN INGLÉS, pero lo subo a mediafire porque no lo encuentro en T! y pienso que puede ser útil para muchos. 

Breve Biografía de Benoit Mandelbrot: 

Benoît Mandelbrot (Varsovia, Polonia, 20 de noviembre de 1924 – Cambridge, Estados

Unidos, 14 de octubre de 2010 ) fue un matemático conocido por sus trabajos sobre los

fractales. Es considerado el principal responsable del auge de este dominio de las

matemáticas desde el inicio de los años setenta, y del interés creciente del público. En

efecto, supo utilizar la herramienta que se estaba popularizando en ésta época - el

ordenador - para trazar los más conocidos ejemplos de geometría fractal: el conjunto de

Mandelbrot por supuesto, así como los conjuntos de Julia descubiertos por Gaston Julia

quien inventó las matemáticas de los fractales, desarrollados luego por Mandelbrot. 

Fuente: WIKIPEDIA

Descarga el libro aquí: 

 

https://dl.dropboxusercontent.com/u/37814585/Mandelbrot%2C%20B.%2C

%20The%20Fractal%20Geometry%20of%20Nature

%20%280716711869%2C%201982%29.pdf 

 

Fractal is a word invented by Mandelbrot to bring together under one heading a large class of

objects that have played ... an historical role ... in the development of pure mathematics. A great

revolution of ideas separates the classical mathematics of the 19th century from the modern

mathematics of the 20th. Classical mathematics had its roots in the regular geometric structures of

Euclid and the continuously evolving dynamics of Newton. Modern mathematics began with

Cantor's set theory and Peano's space filling curve. Historically, the revolution was forced by the

discovery of mathematical structures that did not fit the patterns of Euclid and Newton. These new

structures were regarded ... as ... 'pathological,' ... as a 'gallery of monsters,' akin to the cubist

painting and atonal music that were upsetting the established standards of taste in the arts at about

the same time. The mathematicians who created the monsters regarded them as important in

showing that the world of pure mathematics contains a richness of possibilities going far beyond the

simple structures that they saw in nature. Twentieth century mathematics flowered in the belief that

it had transcended completely the limitations imposed by its natural origins. Now, as Mandelbrot

points out ... Nature has played a joke on the mathematicians. The 19th century mathematicians

may have been lacking in imagination but Nature was not. The same pathological structures that

the mathematicians invented to break loose from 19th century naturalism turns out to be inherent in

familiar objects all around us." ---- Freeman Dyson, "Characterising Irregularity", Science, May

1978 

Fuente: http://www.mountainman.com.au/fractal_00.htm 

 

Fuentes: