Biblioteca de Matemàtiques i Estadística
Carlos Santamarina Macho- E.T.S. Arquitectura Universidad de Valladolid
II Concurso Nacional de Imágenes Fractales
¿Por qué a menudo se describe la geometría como algo "frío" y "seco"? Una de las razones es su incapacidad de describir la forma de una nube, una montaña, una costa o un árbol. Ni las nubes son esféricas, ni las montañas cónicas, ni las costas circulares, ni la corteza es suave, ni tampoco el rayo rectilíneo.
En términos más generales, creo que muchas formas naturales son tan irregulares y fragmentadas que, en comparación con Euclides, la naturaleza no sólo presenta un grado superior de complejidad, sino que ésta se da a un nivel completamente diferente. El número de escalas de longitud de las distintas formas naturales es, a efectos prácticos, infinito.
La existencia de estas formas representa un desafío: el estudio de las formas que Euclides descarta por “informes”, la investigación de la morfología de lo “amorfo”. Los matemáticos, sin embargo, han desdeñado este desafío y, cada vez más, han optado por huir de lo natural, ideando teorías que nada tienen que ver con aquello que podemos ver o sentir.
En respuesta a este desafío, concebí y desarrollé una nueva geometría de la naturaleza y empecé a usarla en una serie de campos. Permite describir muchas de las formas irregulares y fragmentadas que nos rodean, dando lugar a teorías hechas y derechas, identificando una serie de formas que llamo fractales . Las más útiles implican azar, y tanto sus irregularidades son estadísticas. Las formas que describo aquí tienden a ser, también, escalantes , es decir su grado de irregularidad y/o fragmentación es idéntico a todas las escalas. El concepto de dimensión fractal (de Hausdorff) tiene un papel central en esta obra.
Algunos conjuntos fractales son curvas o superficies, otros “polvos” inconexos, y también los hay con formas tan disparatadas que no he encontrado, ni en las ciencias ni en las artes, palabras que los describieran bien. El lector puede hacerse una idea de ello ahora mismo con sólo echar una mirada rápida a las ilustraciones del libro.
Aunque muchas de estas ilustraciones representan formas que nunca antes habían sido consideradas, otras representan, en varios casos por vez primera, construcciones ya conocida. En efecto, la geometría fractal tal como data en 1975, pero muchos de sus útiles y conceptos son anteriores, y aparecieron para satisfacer objetivos muy distintos de los míos. Mediante esas piedras antiguas encajadas en una estructura recien construida, la geometría fratal pudo “tomar prestada” una base excepcionalmente rigurosa, y pronto planteó preguntas nuevas y compulsivas en el terreno de la matemática.
Mandelbrot, Benoit. Los objetos fractales . Barcelona, Tusquets, 1997.
Boston (Mass.) [etc.] : Academic Press, cop. 1993.
Cambridge [etc.] : Cambridge University Press, 1989.
Statistical mechanics and fractals / R.L. Dobrushin, S. Kusukoa.
Berlin [etc.] : Springer-Verlag, cop. 1993.
Berlin [etc.] : Springer, cop. 1994.
Fractals in biology and medicine / Gabriele A. Losa ... [et al.]
Basel [etc.] : Birkhäuser, 1998.
Berlin [etc.] : Springer, 1994.
Berlin, [etc.] : Springer-Verlag, cop. 1992.
Gnomon : from pharaohs to fractals / Midhat J. Gazalé
Princeton : Princeton University Press, cop. 1999.
New York : Springer, cop. 2006.
Chaos, fractals, and noise : stochastic aspects of dynamics / Andrzej Lasota, Michael C. Mackey.
New York [etc.] : Springer, cop. 1994.
New York [etc.] : Springer, cop. 1997.
Geometry of sets and measures in euclidean spaces : fractals and rectifiability / Pertti Mattila.
Cambridge [etc.] : Cambridge University Press, 1995.
Fractals, scaling and growth far from equilibrium / Paul Meakin.
Cambridge [etc.] : Cambridge University Press, 1998.
Fractals for the classroom / Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe.
New York : Springer, cop. 1992.
Chaos and fractals : new frontiers of science / Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe.
New York : Springer-Verlag, cop. 1992.
The Beauty of fractals : images of complex dynamical systems / H.-O. Peitgen, P.H. Richter.
Berlin [etc.] : Springer-Verlag, 1986.
New York : Springer-Verlag, c1989.
Berlin [etc.] : Springer, cop. 2000.
Hackensack, NJ : World Scientific, cop. 2008.
Fractals and Hyperspaces / by Keith R. Wicks.
Berlin, Heidelberg : Springer Berlin Heidelberg, 1991.
Signal processing with fractals : a wavelet-based approach / Gregory W. Wornell.
Upper Sadalle River, [New Jersey] : Prentice Hall, cop. 1996.
NOTA: Definicions extretes de la base de dades OXFORD REFERENCE ONLINE CORE COLLECTION .
Attractor The set of points in phase space to which the representative point of a dissipative system (i.e. one with internal friction) tends as the system evolves. The attractor can be: a single point; a closed curve (a limit cycle ), which describes a system with periodic behaviour; or a fractal (or strange attractor ), in which case the system exhibits chaos .
Cantor set Take the closed interval [0, 1]. Remove the open interval that forms the middle third, that is, the open interval ( 1 / 3 , 2 / 3 ). From each of the remaining intervals again remove the open interval that forms the middle third. The Cantor set is the set that remains when this process is continued indefinitely. It consists of those real numbers whose ternary representation (the representation of a number to base 3) (0. d 1 d 2 d 3 … ) 3 has each ternary digit d i equal to either 0 or 2.
Chaos Unpredictable and seemingly random behaviour occurring in a system that should be governed by deterministic laws. In such systems, the equations that describe the way the system changes with time are nonlinear and involve several variables. Consequently, they are very sensitive to the initial conditions, and a very small initial difference may make an enormous change to the future state of the system. Originally, the theory was introduced to describe unpredictability in meteorology, as exemplified by the butterfly effect . It has been suggested that the dynamical equations governing the weather are so sensitive to the initial data that whether or not a butterfly flaps its wings in one part of the world may make the difference between a tornado occurring or not occuring in some other part of the world. Chaos theory has subsequently been extended to other branches of science; for example to turbulent flow, planetary dynamics, and electrical oscillations in physics, and to combustion processes and oscillating reactions in chemistry. See also attractor ; fractal .
Fractal Geometrical figure in which an identical motif is repeated on a reducing scale; the figure is ‘self-similar'. Coined by Benoit Mandelbrot , fractal geometry is closely associated with chaos theory . Fractal objects in nature include shells, cauliflowers, mountains and clouds. Fractals are also produced mathematically in computer graphics.
Fractal A curve or surface generated by a process involving successive subdivision. For example, a snowflake curve can be produced by starting with an equilateral triangle and dividing each side into three segments. The middle segments are then replaced by two equal segments, which would form the sides of a smaller equilateral triangle. This gives a 12-sided star-shaped figure. The next stage is to subdivide each of the sides of this figure in the same way, and so on. The result is a developing figure that resembles a snowflake. In the limit, this figure has ‘fractional dimension' - i.e. a dimension between that of a line (1) and a surface (2); the dimension of the snowflake curve is 1.26. The study of this type of self-similarity in figures is used in certain branches of physics - for example, crystal growth. Fractals are also important in chaos theory and in computer graphics. See also Mandelbrot set .
Fractal A set of points whose fractal dimension is not an integer or, loosely, any set of similar complexity. Fractals are typically sets with infinitely complex structure and usually possess some measure of self-similarity, whereby any part of the set contains within it a scaled-down version of the whole set. Examples are the Cantor set and the Koch curve .
Fractal A set whose Hausdorff -Besicovitch dimension strictly exceeds its topological dimension. Intuitively, a fractal is a set which at all magnifications reveals a set that is exactly the same (self-similar). Such sets can be generated by the repeated application of some collection of maps. The term is generally associated with Benoit Mandelbrot and appeared in the literature in the late 1970s. Many naturally occurring objects, such as trees, coastlines, and clouds, are considered to have fractal properties, hence their interest to computer graphics.
Fractal dimension One of the many extensions of the notion of dimension to objects for which the traditional concept of dimension is not appropriate. The fractal dimension may have a non-integer value. The Koch curve has dimension ln 4/ln 3 ˜ 1.26. Being between 1 and 2, this reflects the fact that the set is, as it were, too ‘thick' to count as a curve and too ‘thin' to count as an area. The Cantor set has dimension ln 2/ln 3. Fractal dimension has found many practical applications in the analysis of chaotic or noisy processes ( see chaos ).
Fractal image compression A lossy image compression technique based on splitting an image into parts that can each be represented by fractals . The representation of an image by a fractal consists of finding an image transformation that, when applied iteratively to any initial image at the decoder, produces a sequence of images that converges to a fractal approximation of the original. Essentially the encoding of the image is then an encoding of the transformation for each part of the image parts, and a description of the decomposition into parts. Fractal compression is computationally intensive. It has the ability to render images that appear lossless at one extreme and on the other hand the compressed images can be very small in size while still producing recognizable images.
Koch curve Take an equilateral triangle as shown in the first diagram. Replace the middle third of each side by two sides of an equilateral triangle pointing outwards. This forms a six-pointed star, as shown in the second diagram. Repeat this construction to obtain the figure shown in the third diagram. The Koch curve , named after the Swedish mathematician Helge von Koch (1870–1924), is the curve obtained when this process is continued indefinitely. The interior of the curve has finite area, but the curve has infinite length.
Mandelbrot set A fractal that produces complex self-similar patterns. In mathematical terms, it is the set of values of c that make the series z n + 1 = ( z n ) 2 + c converge, where c and z are complex numbers and z begins at the origin (0,0). It was discovered by and named after the Polish-born French mathematician Benoit Mandelbrot (1924– ).
Peano curve Take the diagonal of a square, as shown on the left in the diagram. Replace this by the nine diagonals of smaller squares (drawn here in a way that indicates the order in which the diagonals are to be traced out). Now replace each straight section of this by the nine diagonals of even smaller squares, to obtain the result shown on the right. The Peano curve is the curve obtained when this process is continued indefinitely. It has the remarkable property that it passes through every point of the square, and it is therefore described as space-filling . Any similar space-filling curve constructed in the same kind of way may also be called a Peano curve .
Self-similarity See fractal .
Snowflake curve An example of a curve having fractal dimension. Starting with an equilateral triangle, the middle third, PQ , say, of a side is replaced by the two lines PR and RP so that P , Q , and R form the vertices of a smaller equilateral triangle, with R outside the original enclosed region. This process is repeatedly applied to each line segment. The resulting ‘curve' is the snowflake curve and has infinite length but encloses a finite area. Its fractal dimension is defined to be ln 4 / ln 3, since each ‘edge' of the curve contains four copies each of 1 / 3 size. Snowflake curve. This is an example of a curve with fractal dimension [inequality] 1.
Strange attractor See attractor .
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