Mandelbrot Develops Non-Euclidean Fractal Measures

Date 1966-1975

Benoit B. Mandelbrot developed a simple quantitative model of complex spatial processes that showed scale invariance. Through Mandelbrot’s model, fractal geometry has become, among other things, a method for quantifying the visual components of life.

Locale New York, New York

Key Figures

• Benoit B. Mandelbrot (b. 1924), Polish-born mathematician who pioneered fractals

Summary of Event

Sometime around 1900, the logical, interpretative, and metamathematical problems associated with reconciling Euclidean and non-Euclidean geometries, as well as geometrical with newer Cantorian set-theoretical notions of measure and dimension, were first seriously recognized by Émile Borel and Henri Léon Lebesgue. From 1910 to 1925, French mathematicians Pierre Fatuo and Gaston Julia independently investigated several new set-theoretic approaches to the study of complex two- and three-dimensional spatial distributions using the iterative method of rational functions. These ideas were examined further by French mathematician Constantin Carathéodory in 1914, who sought to define a generalized “length” and “area” for non-Euclidean and other nonstandard shapes, without reliance on traditional coordinate axes.

As Benoit B. Mandelbrot and others recount, the next basic step in further developing these ideas was taken by German mathematician Felix Hausdorff, noted for his monograph Grundzuge der Mengenlehre (1914; Foundations of Set Theory, 1962). In 1919, as an extension of his set-theoretic studies, Hausdorff proposed that “D,” the dimensional order, could be an arbitrary nonnegative integer or noninteger (fractional) and still define a spatial dimension.

Mandelbrot states that his first interest in spatial scaling was triggered, between 1951 and 1961, by empirical regularities governing word frequencies, and later by time variations of commodity prices on competitive markets. A common theme in Mandelbrot’s observations of such phenomena was the fact that many such time series are frequently very discontinuous, as well as being “cyclic,” yet nonperiodic, so that straightforward applications of Fourier methods of harmonic analysis were not possible. In 1961, Mandelbrot further examined scaling of statistical noise in electronic and other systems, as well as so-called Brownian, or 1/f, noise.

The original method of analysis for fractionality was developed previously by the English physicist-mathematician Lewis Fry Richardson in 1961, in his rather unlikely titled publication, “The Problem of Contiguity: An Appendix of Statistics of Deadly Quarrels.” In brief, the length of curves of a coastline, river, mountain range, fault, or international boundary is proposed to be measured by walking a map divider along the curve, with divider points set at each particular spacing. The spacing is changed and the measurement repeated. The total measured line length becomes greater at smaller point spacings (sampling intervals) since more fine detail is encountered, taking progressively smaller and smaller wiggles into account that were missed at larger coarser sample spacings. Richardson first determined that, for a variety of natural boundaries, a log-log (power law) plot of total length versus unit length closely approximates a straight line with a single slope.

To Richardson, the power law exponent in question was merely a value with no special significance. Mandelbrot, having come upon Richardson’s study after reviewing scaling of economic time series, was the first to suggest that the exponent—even though not an integer—could be interpreted meaningfully as a fractional dimension. The basis for this conclusion was that spatial features of natural coastlines are scale invariant under positive or negative magnification, such that coastline photographs taken from altitudes of 10 meters, 1,000 meters, or 100 kilometers are intrinsically indistinguishable, unless a specific reference scale or legend is introduced.

Many subsequent studies in topographic terrain analysis, quantitative geomorphology, engineering materials, and earthquake fracture suggested that self-similarity (scale-invariant shape) apparently occurs in many different natural patterns, where additional finer features are superposed on larger scales. Mandelbrot subsequently proposed to use the formal set-theoretic Hausdorff-Bescovitch dimension. Fractional self-similarity thus clearly manifests itself in a power law. Because lineal and areal fractional measure is undefined in the classical sense, the Hausdorff-Bescovitch, or fractional, dimension is not a spatial or a Euclidean dimension, but a more general topological dimension.

Another basic attribute of fractional measures of coastlines, topographic profiles, cloud shapes, and the like is their nondifferentiability. That is, for the coastline example, although it seems possible to draw tangents to points on the coastline observed from a given height, as one gets closer, more details become visible, and the concept of unique and well-defined tangents becomes meaningless. Nevertheless, the fractional dimension quantifies many characteristics in the spatial variation and irregularity of such a curve.

The term “fractal” was first coined by Mandelbrot in the original French-language edition of his monograph Les Objets fractals (1975; Fractals, 1977). “Fractal” derives from the Latin adjectival root fractus, the same root as in fraction, fracture, and fragment. Notwithstanding, a fractal is not a physical theory or datum, but a mathematical concept or model. Mandelbrot subsequently reinterpreted a number of what he called pre- or latent-fractal data sets, such as the time series of stream profiles in the states of Virginia and Maryland, or the general multiscale roughness of particular land and undersea mountain ranges.

One of the earliest “fractals to be” was the so-called von Koch snowflake, first discussed as a curiosity in an Italian mathematics journal of 1904. Here, a simple line segment is divided into thirds, the middle segment replaced by two equal segments forming part of an equilateral triangle. Next, each of these four segments is replaced by four new segments of length one-third the original length. This procedure (iteratively) repeated yields a well-known six-pointed star snowflake, having similar details over all length scales.

Mandelbrot’s first steps toward explicitly developing a systematic theory of fractal geometry were taken at the T. J. Watson IBM Research Center in central New York State. From 1966 to 1975, Mandelbrot worked in conjunction with hydrologists, meteorologists, statistical physicists, geologists, and many others. Further close relations between fractal measure and traditional Fourier (transform) analysis were underscored in 1968 when Mandelbrot developed a power-law form frequency spectrum for hydrologic time series.

Perhaps the most successful liaison of fractal geometry with science came from Mandelbrot’s studies of spatial patterns in turbulence. In 1963, a physicist first derived a complete set of nonlinear differential equations to approximate thermal convective motion in a fluid. This set of equations was recognized subsequently to be the first exhibiting so-called chaotic behavior, wherein even infinitesimal variations of the initial conditions governing these equations result in major differences in their subsequent mathematical solutions. What Mandelbrot discovered was an important and previously unrecognized feature of many such dynamical systems: their apparent scale invariance. Here, it was shown that fractal measures can be used frequently to describe not only static geometrical patterns but also more dynamic system properties.

Significance

Other phenomena with diffusion-controlled, or random-walk, behavior include two-phase fluid flow, as when a nonviscous fluid such as water is pumped into a viscous fluid like oil, in a porous parent medium such as rock (this is done routinely to enhance secondary petroleum recovery from oil wells). The water typically breaks into a complex branching pattern, the oil flowing more easily away governed by rock pore geometry. Because the thermo-hydro-dynamic equations rigorously describing viscous fluid flow in porous media are extremely complex to solve in full form, it is necessary to introduce accurate simplifications that are physically realistic. A fractal model has, in many cases, been shown to provide a good representation of fluid dynamics in porous media. Another, lesser known but significant, example of fractal applications arises in the areas of signal detection in digital signal and image processing, as in radar, acoustics, and seismology.

Other examples suggest that fractal geometry and dynamics may apply with sufficient accuracy to a wide variety of physical, chemical, and biological phenomena. If this is true, then many currently intractable equations describing heat transfer, as well as wave propagation and scattering through complex rough-surfaced media, may find solutions of sufficient accuracy to match observed data. As noted, however, by several mathematical geologists, geophysicists, and others who have investigated power-law and other spatial pattern analyses, there are several practical problems, as well as possible problems in principle, in the more universal and rigorous applications of fractals.

Numerous contributions to several interdisciplinary and topical conferences, as well as collaborative research, have ensured the further consideration and applications of deterministic and stochastic fractal characterizations. A characteristic of Mandelbrot’s fractals research program is his continuous careful search for theoretical and empirical results that fit well with the basic fractal concept. As evident from his general audience publications in 1977, 1982, and 1990, Mandelbrot’s rather broad aesthetic feeling, and mixing of theory with applications in conveying new ideas, has played an important part in wider recognition and consideration of fractal methods.

Mandelbrot’s work has opened the way to a wider recognition and use of other, lesser known mathematical notions, such as “fuzzy sets,” “generalized regional variables,” and the “catastrophe theory.” Considered as an episode in the history of science, the existence of a real physical example endowed with certain mathematical properties is in no way a precondition for the discovery or use of the mathematics in question. Nevertheless, in the case of fractals, it was not primarily abstract contemplation or calculation but specific considerations of concrete examples of scaling that led to the “discovery” of fractals and their wider use.

Occasions where new ideas and methods enter numerous sciences by way of obscure subbranches of mathematics are rare in the history of the natural sciences. Likewise uncommon is a new discipline (particularly of wide applications scope) whose foundations are established largely without direct support of previously published work. The discovery and further development of the theory and applications of fractals is a curious and continuing example of such historical anomalies.

Bibliography

Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Cambridge, Mass.: Harvard University Press, 1979. Offers invaluable conceptual and methodological background to the set-theoretic history of fractional measure.

Mandelbrot, B. B. Fractals and Multifractals: Noise, Turbulence, and Galaxies. New York: Springer-Verlag, 1990. Mandelbrot’s most comprehensive and rigorous treatment of fractals.

‗‗‗‗‗‗‗. Fractals: Form, Chance, and Dimension. San Francisco: W. H. Freeman, 1977. The first English-language general-audience monograph on the history and concepts of fractional dimension.

Peitgen, Heinz-Otto, and Dietmar Saupe, eds. The Science of Fractal Images. New York: Springer-Verlag, 1988. A brief review of key concepts. Outlines computer programs to generate a catalog of fractal graphics.

Scholz, Christopher, and Benoit B. Mandelbrot. Fractals in Geophysics. Boston: Kirkäuser, 1989. A collection of papers for and against the utility of fractals in representing multiscale topographic relief.

Stanley, H. Eugue, and Nicole Ostrowsky, eds. On Growth and Form: Fractal and Non-Fractal Patterns in Physics. Dordrecht, the Netherlands: M. Nijhoff, 1986. The first conference proceedings dedicated to documenting fractal applications in solid-state physics, diffusion, percolation, critical point phenomena, and Brownian motion.

Stewart, Ian, ed. The Colours of Infinity: The Beauty and Power of Fractals. Bath, England: Clear Books, 2004. Perhaps the most straightforward, nontechnical outline and discussion of what Mandelbrot came to call fractals. Highly recommended for all readers at all levels. Includes a stunning DVD, introduced by writer Arthur C. Clarke, of the British television documentary that helped to popularize the subject.