THE FRACTAL GEOMETRY OF NATURE

Benoit B. Mandelbrot

Introduction

THEME

Why is geometry often described as "cold" and "dry?" One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

More generally, I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid-a term used in this work to denote all of standard geometry-Nature exhibits not simply a higher degree but an altogether different level of complexity. The number of distinct scales of length of natural patterns is for all practical purposes infinite.

The existence of these patterns challenges us to study those forms that Euclid leaves aside as being "formless," to investigate the morphology of the "amorphous." Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel.

Responding to this challenge, I conceived and developed a new geometry of nature and implemented its use in a number of diverse fields. It describes many of the irregular and fragmented patterns around us, and leads to full-fledged theories, by identifying a family of shapes I call fractals. The most useful fractals involve chance and both their regularities and their irregularities are statistical. Also, the shapes described here tend to be scaling, implying that the degree of their irregularity and/or fragmentation is identical at all scales. The concept of fractal (Hausdorff) dimension plays a central role in this work.

Some fractal sets are curves or surfaces, others are disconnected "dusts," and yet others are so oddly shaped that there are no good terms for them in either the sciences or the arts. The reader is urged to sample them now, by browsing through the book's illustrations.

Many of these illustrations are of shapes that had never been considered previously, but others represent known constructs, often for the first time. Indeed, while fractal geometry as such dates from 1975, many of its tools and concepts had been previously developed, for diverse purposes altogether different from mine. Through old stones inserted in the newly built structure, fractal geometry was able to "borrow" exceptionally extensive rigorous foundations, and soon led to many compelling new questions in mathematics.

Nevertheless, this work pursues neither abstraction nor generality for its own sake, and is neither a textbook nor a treatise in mathematics. Despite its length, I describe it as a scientific Essay because it is written from a personal point of view and without attempting completeness. Also, like many Essays, it tends to digressions and interruptions.

This informality should help the reader avoid the portions lying outside his interest or beyond his competence. There are many mathematically, "easy" portions throughout, especially toward the very end. Browse and skip, at least at first and second reading.

PRESENTATION OF GOALS

This Essay brings together a number of analyses in diverse sciences, and it promotes a new mathematical and philosophical synthesis. Thus, it serves as both a casebook and a manifesto. Furthermore, it reveals a totally new world of plastic beauty.

A SCIENTIFIC CASEBOOK

Physicians and lawyers use "casebook" to denote a compilation concerning actual cases linked by a common theme. This term has no counterpart in science, and I suggest we appropriate it. The major cases require repeated attention, but less important cases also deserve comment; often, their discussion is shortened by the availability of "precedents."

One case study concerns the widely known application of widely known mathematics to a widely known natural problem: Wiener's geometric model of physical Brownian motion. Surprisingly, we encounter no fresh direct application of Wiener's process, which suggests that, among the phenomena of higher complexity with which we deal, Brownian motion is a special case, an exceptionally simple and unstructured one. Nevertheless, it is included because many useful fractals are careful modifications of Brownian motion.

The other case studies report primarily upon my own work, its pre-fractal antecedents, and its extensions due to scholars who reacted to this Essay's 1975 and 1977 predecessors. Some cases relate to the highly visible worlds of mountains and the like, thus fulfilling at long last the promise of the term geometry. But other cases concern submicroscopic assemblies, the prime object of physics.

The substantive topic is occasionally esoteric. In other instances, the topic is a familiar one, but its geometric aspects had not been attacked adequately. One is reminded on this account of Poincare's remark that there are questions that one chooses to ask and other questions that ask themselves. And a question that had long asked itself without response tends to be abandoned to children.

Due to this difficulty, my previous Essays stressed relentlessly the fact that the fractal approach is both effective and "natural." Not only should it not be resisted, but one ought to wonder how one could have gone so long without it. Also, in order to avoid needless controversy, these earlier texts minimized the discontinuities between exposition of standard and other published material, exposition with a new twist, and presentation of my own ideas and results. In the present Essay, to the contrary, I am precise in claiming credit.

Most emphatically, I do not consider the fractal point of view as a panacea, and each case analysis should be assessed by the criteria holding in its field, that is, mostly upon the basis of its powers of organization, explanation, and prediction, and not as example of a mathematical structure. Since each case study must be cut short before it becomes truly technical, the reader is referred elsewhere for detailed developments. As a result (to echo d'Arcy Thompson 1917), this Essay is preface from beginning to end. Any specialist who expects more will be disappointed.

A MANIFESTO: THERE IS A FRACTAL
FACE TO THE GEOMETRY OF NATURE

Now, the reason for bringing these prefaces together is that each helps one to understand the others because they share a common mathematical structure. F. J. Dyson has given an eloquent summary of this theme of mine. "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,' kin to the cubist painting and atonal music that were upsetting 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 turn out to be inherent in familiar objects all around us."

In brief, I have confirmed Blaise Pascal's observation that imagination tires before Nature. ("L'imagination se lassera plutot de concevoir que la nature de fournir.")

Nevertheless, fractal geometry is not a straight "application" of 20th century mathematics. It is a new branch born belatedly of the crisis of mathematics that started when duBois Reymond 1875 first reported on a continuous nondifferentiable function constructed by Weierstrass (Chapters 3, 39, and 41). The crisis lasted approximately to 1925, major actors being Cantor, Peano, Lebesgue, and Hausdorff. These names, and those of Besicovitch, Bolzano, Cesaro, Koch, Osgood, Sierpinski, and Urysohn, are not ordinarily encountered in the empirical study of Nature, but I claim that the impact of the work of these giants far transcends its intended scope.

I show that behind their very wildest creations, and unknown to them and to several generations of followers, lie worlds of interest to all those who celebrate Nature by trying to imitate it.

Once again, we are surprised by what several past occurrences should have led us to expect, that "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" (Wigner 1960).

MATHEMATICS, NATURE, ESTHETICS

In addition, fractal geometry reveals that some of the most austerely formal chapters of mathematics had a hidden face: a world of pure plastic beauty unsuspected till now.

"FRACTAL" AND OTHER NEOLOGISMS

There is a saying in Latin that "to name is to know:" Nomen est numen. Until I took up their study, the sets alluded to in the preceding sections were not important enough to require a term to denote them. However, as the classical monsters were defanged and harnessed through my efforts, and as many new 'monsters" began to arise, the need for a term became increasingly apparent. It became acute when the first predecessor of this Essay had to be given a title.

I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means "to break:" to create irregular fragments. It is therefore sensible-and how appropriate for our needs!-that, in addition to "fragmented" (as infraction or refraction), fractus should also mean "irregular," both meanings being preserved in fragment.

The proper pronunciation is frac'tal, the stress being placed as infraction.

The combination fracial set will be defined rigorously, but the combination natural fractal will serve loosely to designate a natural pattern that is usefully representable by a fractal set. For example, Brownian curves are fractal sets, and physical Brownian motion is a natural fractal.

(Since algebra derives from the Arabic jabara = to bind together, fractal and algebra are etymological opposites!)

More generally, in my travels through newly opened or newly settled territory, I was often moved to exert the right of naming its landmarks. Usually, to coin a careful neologism seemed better than to add a new wrinkle to an already overused term.

And one must remember that a word's common meaning is often so entrenched, that it is not erased by any amount of redefinition. As Voltaire noted in 1730, "if Newton had not used the word attraction, everyone in [the French] Academy would have opened his eyes to the light; but unfortunately he used in London a word to which an idea of ridicule was attached in Paris." And phrases like "the probability distribution of the Schwartz distribution in space relative to the distribution of galaxies" are dreadful.

The terms coined in this Essay avoid this pitfall by tapping underutilized Latin or Greek roots, like trema, and the rarely borrowed robust vocabularies of the shop, the home, and the farm. Homely names make the monsters easier to tame! For example, I give technical meanings to dust, curd, and whey. I also advocate pertiling for a thorough (= per) form of tiling.

RESTATEMENT OF GOALS

In sum, the present Essay describes the solutions I propose to a host of concrete problems, including very old ones, with the help of mathematics that is, in part, likewise very old, but that (aside from applications to Brownian motion) had never been used in this fashion. The cases this mathematics allows us to tackle, and the extensions these cases require, lay the foundation of a new discipline.

Scientists will (I am sure) be surprised and delighted to find that not a few shapes they had to call grainy, hydralike, in between, pimply, pocky, ramified, seaweedy, strange, tangled, tortuous, wiggly, wispy, wrinkled, and the like, can henceforth be approached in rigorous and vigorous quantitative fashion.

Mathematicians will (I hope) be surprised and delighted to find that sets thus far reputed exceptional (Carleson 1967) should in a sense be the rule, that constructions deemed pathological should evolve naturally from very concrete problems, and that the study of Nature should help solve old problems and yield so many new ones.

Nevertheless, this Essay avoids all purely technical difficulties. It is addressed primarily to a mixed group of scientists. The presentation of each theme begins with concrete and specific cases. The nature of fractals is meant to be gradually discovered by the reader, not revealed in a flash by the author.

And the art can be enjoyed for itself.

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