Fractals in the Natural Sciencesby M. Fleischmann
In the words of B. B. Mandelbrot's contribution to this important collection of original papers, fractal geometry is a "new geometric language, which is geared towards the study of diverse aspects of diverse objects, either mathematical or natural, that are not smooth, but rough and fragmented to the same degree at all scales." This book will be of interest to all
In the words of B. B. Mandelbrot's contribution to this important collection of original papers, fractal geometry is a "new geometric language, which is geared towards the study of diverse aspects of diverse objects, either mathematical or natural, that are not smooth, but rough and fragmented to the same degree at all scales." This book will be of interest to all physical and biological scientists studying these phenomena. It is based on a Royal Society discussion meeting held in 1988.
Originally published in 1990.
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Fractals in the Natural Sciences
By M. Fleischmann, D. J. Tildesley, R. C. Ball
PRINCETON UNIVERSITY PRESSCopyright © 1989 The Royal Society of London
All rights reserved.
Fractal geometry: what is it, and what does it do?
BY B. B. MANDELBROT
Physics Department, IBM T. J. Watson Research Center, Yorktown Heights, New York 10598. U.S.A.
Mathematics Department, Yah University, New Haven, Connecticut 06520, U.S.A.
Fractal geometry is a workable geometric middle ground between the excessive geometric order of Euclid and the geometric chaos of general mathematics. It is based on a form of symmetry that had previously been underused, namely invariance under contraction or dilation. Fractal geometry is conveniently viewed as a language that has proven its value by its uses. Its uses in art and pure mathematics, being without 'practical' application, can be said to be poetic. Its uses in various areas of the study of materials and of other areas of engineering are examples of practical prose. Its uses in physical theory, especially in conjunction with the basic equations of mathematical physics, combine poetry and high prose. Several of the problems that fractal geometry tackles involve old mysteries, some of them already known to primitive man, others mentioned in the Bible, and others familiar to every landscape artist.
FRACTALS PROVIDE A WORKABLE N E W MIDDLE GROUND BETWEEN THE EXCESSIVE GEOMETRIC ORDER OF EUCLID AND THE GEOMETRIC CHAOS OF ROUGHNESS AND FRAGMENTATION
Instead of attempting to introduce and link together the papers that follow in this Discussion Meeting, we prefer to ponder the question, 'What is fractal geometry?' We write primarily for the comparative novice, but have tried to include tidbits for the already informed reader.
Before we tackle what a fractal is, let us ponder what a fractal is not. Take a geometric shape and examine it in increasing detail. That is, take smaller and smaller portions near a point P, and allow every one to be dilated, that is, enlarged to some prescribed overall size. If our shape belongs to standard geometry, it is well known that the enlargements become increasingly smooth. Ultimately, nearly every connected shape is locally linear. One can say, for example, that 'a generic curve is attracted under dilations' to a straight line (thus defining the tangent at the point P). And 'a generic surface is attracted by dilation' to a plane (thus defining the tangent plane at the point P). More generally, one can say that nearly every standard shape's local structure converges under dilation to one of the small number of 'universal attractors'. The term 'attractor' is borrowed from dynamics and probability theory, and the even more grandiose term, 'universal', is borrowed from recent physics. An example of exception to this rule is when P is a double point of a curve ; the curve near P is then attracted to two intersecting lines and has two tangents; but double points are few and far between in standard curves.
Standard geometry and calculus (which is intimately related to it) have long proven to be extraordinarily effective in the sciences. Yet there is no question that Nature fails to be locally linear. Indeed, the shapes of Nature are so varied as to deserve being called 'geometrically chaotic', unless proven otherwise. Unfortunately, 'complete' chaos could not conceivably lead to a science. This is perhaps why many of the oldest concerns of Man, such as concerns with the shapes of mountains, clouds and trees or, with the floods of the Nile, had not led to sciences comparable in effectiveness to the physics of smooth phenomena.
Though the term 'chaos' was not used, one can say that a second kind of chaos became known during the half century, 1875–1925, when mathematicians who were fleeing from concerns with Nature took cognizance of the fact that a geometry shape's roughness may conceivably fail to vanish as the examination becomes more searching. It may conceivably vary endlessly, up and down. The hold of standard geometry was so powerful, however, that the shapes constructed so that they do not reduce locally to straight lines were labelled 'monsters' and 'pathological'.
Between the extremes of linear geometric order and of geometric chaos ruled by 'pathologies', can there be a middle ground of 'organized' or 'orderly' geometric chaos? The author has conceived and outlined such a ground, and gave its study the name of 'fractal geometry', the fuller name being 'fractal geometry of nature and chaos'. It will be argued momentarily that fractal geometry is best viewed as a geometric language, new as of 1975, which incorporates as 'characters' several of the mathematical monsters of 1875–1925, and whose uses have now become so diverse, that it is possible to sort them out as poetry, strictly utilitarian prose and high prose.
FRACTALS ARE CHARACTERIZED BY SO-CALLED 'SYMMETRIES', WHICH ARE INVARIASCES UNDER DILATIONS AND/OR CONTRACTIONS
Broadly speaking, mathematical and natural fractals are shapes whose roughness and fragmentation neither tend to vanish, nor fluctuate up and down, but remain essentially unchanged as one zooms in continually and examination is refined. Hence, the structure of every piece holds the key to the whole structure.
An alternative term is 'self-similar', which has two meanings. One can understand 'similar' as a loose everyday synonym of' analogous'. But there is also the strict textbook sense of 'contracting similarity'. It expresses that each part is a linear geometric reduction of the whole, with the same reduction ratios in all directions. Figure 1 illustrates a standard strictly self-similar fractal, called Sierpinski gasket. In the early days, the resulting terminological ambiguity was acceptable to physicists, because early detailed studies did indeed concentrate on strictly self-similar shapes. However, more recent developments have extended, in particular, to include self-affine shapes, in which the reductions are still linear but the reduction ratios in different directions are different. For example, a relief is nearly self-affine, in the sense that to go from a large piece to a small piece one must contract the horizontal and vertical coordinates in different ratios.
Can the process of taking parts be inverted, by replacing zooming in by zooming away from the object of interest ? Indeed, many fractals, including the strictly self-similar ones, can be extrapolated to become unbounded, and they can remain unchanged when the above process is reversed and the shaped is examined at increasingly rough scales. To gain an idea of the appearance of an extrapolated Sierpinski gasket, it suffices to focus on a very small piece of figure 1, and then to imagine that this piece has been blown up to letter size, and therefore the whole gasket has become so large that its edges are beyond visibility. This illustration shows that each choice of a small piece to focus upon yields a different extrapolate.
When the Sierpinski gasket is constructed as in figure 1, that is, by deleting middle triangles, one sees it has three properties of contracting self-similarity, as pointed out in the caption of figure 1. These properties appear, so to speak, as 'static' and 'after-the-fact', but this is a completely misleading impression. Its prevalence and its being viewed as a flaw came to us as a surprise. Therefore, it is good to stop and show how, knowing the same symmetries, it is easy to reconstruct the gasket 'dynamically' by a stochastic interpretation of a scheme due to J. Hutchinson. The basic principle of this scheme first arose long ago, in the work of Poincaré and Klein, and corresponding illustrations using randomization are found in our book (Mandelbrot 1982). Start with an 'initiator' that is an arbitrary bounded set, for example is a point P0. Denote the three similarities of the gasket by S0, S1 and S2, and denote by k(m) a random sequence of the digits 0, 1 and 2. Then define an 'orbit' as made of the points P1 = Sk(1)(P0), P2 = Sk(2)(P1) and more generally Pj = Sk(j)(Pj-1). One finds that this orbit is 'attracted' to the gasket, and that after a few stages it describes its shape very well.
SURPRISE: SIMPLE RULES CAN GENERATE RICH STRUCTURE
How did fractals come to play their role of 'extracting order out of chaos?' To understand, one must go beyond simple shapes like the gasket or like the other fractals-to-be that mathematicians have first introduced as counter-examples. In these 'old' shapes, indeed, what one gets out follows easily from what has been knowingly put in. The key to fractal geometry's effectiveness resides in a very surprising discovery the author has made, largely thanks to computer graphics.
The algorithms that generate the other fractals are typically so extraordinarily short, as to look positively dumb. This means they must be called 'simple'. Their fractal outputs, to the contrary, often appear to involve structures of great richness.
A priori, one would have expected that the construction of complex shapes would necessitate complex rules. Thus, fractal geometry can be the study of geometric shapes that may seem chaotic, but are in fact perfectly orderly.
Let us, for the sake of contrast, comment on the examples of a related match between mathematics and the computer that arise in areas such as the study of water eddies and wakes. In these examples, the input in terms of reasoning or of number of lines of program is extremely complicated, perhaps more complicated even than the output. Therefore, one may argue that, overall, total complication does not increase in those examples, merely changes over from being purely conceptual to being partly visual. This change-over is very important and very interesting, but fractal geometry gives us something very different.
What is the special feature that makes fractal geometry perform in such unusual manner? The answer is very simple; the algorithm involves 'loops'. That is, the basic instructions are simple, and their effects can be followed easily. But let these simple instructions be followed repeatedly. Unless one deals with the simplest old fractals (Cantor set or Sierpinski gasket), the process of iteration effectively builds up an increasingly complicated transform, whose effects the mind can follow less and less easily. Eventually, one reaches something that is 'qualitatively' different from the original building block. One can say that the situation is a fulfilment of what is general is nothing but a dream: the hope of describing and explaining 'chaotic' nature as the cumulation of many simple steps.
FRACTAL GEOMETRY VIEWED AS A LANGUAGE
'Mathematics is a language.' (Josiah Willard Gibbs (speaking at a Yale faculty meeting ... on elective course requirements).)
'Philosophy is written in this vast book–I mean the Universe–which stands forever open to our gaze, but cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles and other geometrical figures, without which it is humanly impossible to understand a single word of it; without which one wanders in vain through a dark labyrinth.' (Galileo Galilei: Il Saggiatore (The Assayer) 1623).
'The language of mathematics reveals itself unreasonably effective in the natural sciences ..., a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure even though perhaps also to our bafflement, to wide branches of learning.' (Eugene Wigner 1960).
Inspired by the above quotes, the best is to call fractal geometry a new geometric language, which is geared towards the study of diverse aspects of diverse objects, either mathematical or natural, that are not smooth, but rough and fragmented to the same degree at all scales. Its history is interesting and curious. As is clearly indicated by such terms as 'Cantor set', 'Peano curve', 'Sierpinski gasket', etc. ..., several fractal 'characters' date to 1875–1925. They count among the least complex (hence least beautiful), and they have seen previous uses in other languages that have nothing to do with fractal geometry.
However, as a language addressed to its new goals, fractal geometry was born with Mandelbrot (1975), the first edition of our book Les objets fractals.
There is a profound historical irony in the fact that these old 'characters' of the new geometry had been among the 'monsters' to which we have referred earlier. The general monster is not scaling, and the fact that some monsters are scaling had not been singled out, because it was viewed as a 'special' property, therefore one that is not very 'interesting'. In the mathematical culture of the century that ran from 1875 to 1975, special properties did not warrant investigation.
Clearly, the study of order in geometric chaos could not arise as a specialized object of study, and a term to denote it did not become indispensable, until we had performed two tasks, (i) We saw that diverse rough patterns of nature – noise and turbulence and diverse geographical features – are geometrically scaling. Eventually we, and now many others, have identified geometric scaling in many other areas of nature, and then explored its consequences, (ii) We saw that the proper tool to tackle scaling in nature is suggested by some of the old 'monsters'. The old monsters themselves are not realistic models, but the construction of new fractals was immediately spurred by the new need, and more accurate models soon became available.
A language can be appreciated in diverse ways. For fractal geometry, the reasons can be sorted into five categories, artistic, mathematical, 'historical', practical and scientific. Let us discuss them in turn.
THE LANGUAGE OF FRACTAL GEOMETRY OFTEN CREATES PLASTIC BEAUTY
We use the term 'geometry' in a very archaic sense, as involving concrete actual images. Lagrange and Laplace once boasted of the absence of any pictures in their works, and their lead was eventually followed almost universally. Fractal geometry is a reaction against the tide, and a first reason to appreciate fractal geometry, because the 'characters' it adds to the 'alphabet' Galileo had inherited from Euclid, often happen to be intrinsically attractive. Many have promptly been accepted as works of a new form of art. Some are 'representational', in fact are surprisingly realistic 'forgeries' of mountains, clouds or trees, while others are totally unreal and abstract. Yet all strike almost everyone in forceful, almost sensual, fashion. The artist, the child and the 'man in the street' never seem to have seen enough, and they had never expected to receive anything of this sort from mathematics. Neither had the mathematician expected his field to interact with art in this way.
In any event, fractal geometry shows that there is an unexpected parallel to the above classic quote from Wigner. We have been fortunate to witness the revelation of the "'unreasonable' and 'undeserved' effectiveness of mathematics as a source of enjoyable form."
Excerpted from Fractals in the Natural Sciences by M. Fleischmann, D. J. Tildesley, R. C. Ball. Copyright © 1989 The Royal Society of London. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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