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SCRAPBOOK OF SELECTA BOOKS FOREWORDS & REVIEWS
Benoit B. Mandelbrot
September 29, 2006
THIS FILE IS IN FIVE PARTS,
EACH DEVOTED TO ONE OF THE FOLLOWING BOOKS
SE Fractals and Scaling in Finance: Discontinuity, Concentration, Risk.
New York: Springer, 1997, x+551pp.
SFE Fractales, hasard et finance (1959  1997).
Paris: Flammarion (Collection Champs), 1997, 246pp.
SN Multifractals and 1/f Noise : Wild Selfaffinity in Physics.
New York: Springer. 1999, viii + 442 pp.
SH Gaussian SelfAffinity and Fractals: Globality, the Earth, 1/f, and R/S.
New York: Springer. 2002, ix + 654 pp.
SC Fractals and Chaos: the Mandelbrot Set and Beyond.
New York: Springer. 2004, xii + 308 pp.
FRACTALS AND SCALING IN FINANCE
Foreword by Ralph E. Gomory
(President, Sloan Foundation)
In 195961, while the huge Saarinendesigned research laboratory at Yorktown Heights was being built, much of IBM's Research was housed nearby. My group occupied one of the many little houses on the Lamb Estate complex which had been a sanatorium housing wealthy alcoholics.
Even in a Lamb Estate populated exclusively with bright researchoriented people, Benoit always stood out. His thinking was always fresh, and I enjoyed talking with him about any subject, whether technical, political, or historical. He introduced me to the idea that distributions having infinite second moments could be more than a mathematical curiosity an a source of counterexamples.
This was a foretaste of the line of thought that eventually led to fractals and to the notion that major pieces of the physical world could be, and in fact could only be, modeled by distributions and sets that had fractional dimensions. Usually these distributions and sets were known to mathematicians, as they were known to me, as curiosities and counterintuitive examples used to show graduate students the need for rigor in their proofs.
I can remember hearing Benoit assert that daytoday changes of stock prices have an infinite second moment. The consequence was that most of the total price change over a long period was concentrated in a few hectic days of trading and it was there that fortunes were made and lost. He also asserted that the historical data on stock prices supported this view, that as you took longer and longer historical data, the actual second moments did not converge to any finite number.
His thinking about floods was similar.
Benoit's ideas impressed me enormously, but it was hard to get this work recognized. Benoit was an outsider to the substantive fields that his models applied to, for example economics and hydrology, and he received little support from mathematicians who saw only that he was using known techniques. Benoit's contribution was to show that these obscure concepts lie at the roots of a huge range of real world phenomena.
Lack of recognition, however, never daunted Benoit. He stuck to his ideas and worked steadily to develop them and to broaden their range of applicability, showing that one phenomenon after another could be explained by his work. I was very pleased when I was able to get him named an IBM Fellow, and later was successful in nominating him for the Barnard Medal. After that the floodgates of recognition started to open and Benoit today is one of the most visible of scientific figures.
Surely he has earned that visibility, both for his worldchanging work, and for his courage and absolute steadfastness.
Acta Scientiarum Mathematicarum (Szeged, Hungary) % Lszl I. Szab
& A mixture of newlywritten material, old articles and contributions from other authors. The text is centered around 3 successive models of price variation& .
While some of the author s views and hypotheses are debated by some economists, Mandelbrot s original insights have unquestionably contributed in a substantial way to our understanding of the economic world. This book is a timely work in the age of econophysics.
American Mathematical Monthly %April, 1998 % KB
Highlights a new classification of forms of randomness into states that range from mild to wild; a useful classification of prices departures from Brownian motion, into Noah and Joseph effects and their combination; a broad panorama of old and new forms of selfaffine variability. Theme: although prices vary wildly, scaling rules hold ensuring financial charts are examples of fractal shapes.
Computers and Mathematics with Applications %35, 1998(5).
EMSEuropean Mathematical Society Newsletter %
34, (40), 1999 JH % K.
Presents an alternative approach to the analysis of financial data, or more generally, to any data set possessing features like financial time series. Is recommended to any mathematician and/or financial analyst who wishes to learn more about the variety of
alternative models and to avoid using just the classical methods.
The Guardian %
11/12/97 % Clive Davidson
MANDELBROT S ROLLERCOASTER.
THE DISCOVERER OF THE CHAOS THEORY HAS PUBLISHED HIS IDEAS ON WHY STOCK MARKETS CRASH. SCALING CAN PRODUCE WILDLY RANDOM MOVEMENTS Everyone knows that every so often the markets experience swings of mood, when prices jump in a flurry of trading. But classical market theory says they shouldnt. Prices should make small random movements, rather like particles in a solution bombarded by surrounding molecules. If the markets always followed such Brownian movementprices would steadily zigzag their way up or down in response to changing economic conditions.
But according to Benoit Mandelbrot, the mathematician who discovered fractals, we dont have to abandon the notion of randomness to create a model of the markets that more accurately reflects their reality. In his new book,he argues that fractalbased models give a more realistic picture of financial risk.
One of the principles of fractals is that apparently simple processes can generate unexpectedly complicated and structured forms. We see these in nature, from the shape of plants to geological formations.
Mandelbrot demonstrated the process in the seventies, when he used a computer to produce striking and complex images from relatively simple equations that became known as the Mandelbrot Set.
Although Mandelbrot is best known for these images and for his work on fractals in the physical world, he first stumbled on the phenomenon in the financial markets.
In the fifties, he set out to investigate a piece of financial folklore that suggested that if you took away the time scales of a series of graphs of prices plotted over different intervals you could not tell which was which.
This selfsimilarity at different scales is a feature of fractal systems. Mandelbrot went on to show that one of the features of such scaling systems isthat the changes in such systems are not evenly spread over time but tend to happen in concentrated bursts.A system based on the principles of scaling can produce what Mandelbrot calls wildly random movements, just like the price crash and recovery on October 23.
Financial professionals are most comfortable when market conditions are mild, with small fluctuations in prices. They tend to supplement their Brownian movement models with stress tests, in which they look at what would happen to their portfolios if there was a rerun of the crash of 1987 and so on. Mandelbrots ideas offer a way to build market models that include periods of calm as well as price hikes, crashes and the like.
One of the problems for Mandelbrot in the sixties was the lack of market data, computers and statistical tools. So it was difficult to test his assertions and they were largely ignored.
But two decades later, things had changed.
Mandelbrots theories no longer seem so wild to the financial industry.
IBM Research % November 3, 1997
After a long detour through the rest of the universe, Benoit Mandelbrot s exploration of fractal phenomena has come back to its roots. While it was not until 1975 that he coined the term fractal to refer to mathematical and natural objects characterized by the same extreme degree of irregularity at all scales the underlying ideas had been germinating for much longer. His just published book itself consists of old and new material, in roughly equal parts, including his reprints of articles on finance and economics from 1960 to 1973. In a long, multichapter introduction, Mandelbrot places the evolution of his work in context and explains the new picture of economic phenomena that his ideas entail
As in his other excursions into fields as diverse as condensedmatter physics, mathematics, linguistics, geophysics, fluid dynamics and astronomy to name a few Mandelbrot brought to the study of finance and economics a gift for geometric insight and a capacity to seize on and synthesize ideas that others had either overlooked or failed to see could be applied in a novel context.
At the heart of Mandelbrots approach to economics is a contrast he draws between different states of randomness. From his viewpoint, the randomness dealt with in traditional physics and used by Bachelier in his Brownianmotion, or randomwalk, model of price variations is mild, whereas financial reality is characterized by the state of wild randomness. Thus, he argues, there is no underlying equilibrium whose fluctuations average out; rather, price changes experience cycles of turbulent behavior.
Yet, underlying this extreme randomness are invariance principles arising from a generating process that remains constant in time. The result is that a graph of price changes is invariant in a statistical sense under displacements along the time axis and under change of scale. Such scaling, or selfaffinity (a notion close to selfsimilarity), is, of course, the telltale sign of a fractal.
Mandelbrots goal in creating economic models was to obtain some degree of understanding of phenomena that seemed impervious to mathematical description. By showing that the wild randomness of the data can be modeled more accurately than previously believed in a way that does not depend on a variety of ad hoc fixes, Mandelbrot has also produced practical tools to evaluate the inherent risks of financial trading. The search for understanding must continue, says Mandelbrot, but financial engineering cannot wait for full explanation. Meanwhile, the increased breadth, depth, and accessibility of Mandelbrots ideas will undoubtedly spur new efforts in a field that affects us all.
International Statistical Institute
Short Book Reviews %
P.A.L. Embrechts (ETH, Zrich, CH)
ITW Nieuws (Niederlande) %
Jaan van Oosten
Jahresbericht der Deutscher Mathematiker Vereinigung % 101(2)
M. Schweizer (Technische U., Berlin DE)
As a partial collection of selected papers, the book has certainly quite some historical value and interest. It also presents a remarkable panorama of ideas and clearly shows what Paul Samuelson called Mandelbrots incorrigibly original mind. In other aspects, however, I found some deficiencies. Most important among these is probably the lack of a balanced presentation. While some of the personal comments are entertaining, one misses at least an overview of other work that has been done in the area of Mandelbrots research in finance. There is hardly any mention of the evolution between the sixties and today, and there is no effort to provide a perspective of the field as a whole. As a consequence, some comparisons are quite clearly biased and also not up to date
In summary, this book is a useful collection of Mandelbrots most important papers on modeling in finance, supplemented by some more recent ideas on the role of scaling rules in that field.
Journal of Economic Literature and eJEL,JEL on CD, and EconLit 38 (3)
Elaborates on the tendency of stock market price changes to concentrate in turbulent
periods in a series of newly written essays followed by reprinted papers that give historical depth and add technical detail. Discusses discontinuity and scaling, their scope and likely
limitations; sources of inspiration and the historical background; states of randomness and concentration in the short, medium, and long run; selfsimilarity and panorama of selfaffinity; proportional growth and other explanations of scaling; and a case against the lognormal distribution. Three sections of reprinted papers examine personal incomes and firm sizes; test or comment on the author's 1963 model of price variation; and present steps beyond the 1963 model.
Journal of Economic Literature % XXXIX June 2001 % Philip E. Mirowski (U. of Notre Dame)
Benoit Mandelbrot is an imaginative scholar, but one whom those equipped with firm disciplinary loyalties have found it a struggle to understand. This has been a problem across the disciplinary spectrum, although in economics it has assumed one of two forms: there are the neoclassical finance theorists, who argue against what they have perceived as his notions precisely because they clash with received microeconomic theory (although empirical controversies have also played a role); and then there are the selfdesignated "econophysicists," refugees from the natural sciences with no particular doctrinal orthodoxies to defend, who have been attracted to his work precisely because of its undeniable influence in the physics of turbulence, diffusion processes, semiconductors, and elsewhere. It used to be that Mandelbrot would confound both his supporters and detractors by insisting upon a third stance, which went roughly: research should be guided by the precept that the geometric character of any given phenomenon should be the primary heuristic, combined with a fearless acceptance of indeterminism; no inquiry should be prematurely stifled by either entrenched dogmas or by illconceived physics envy. In economics, this amounted to repeated exercises arguing that empirical price distributions were fattailed, exhibiting long dependence, and altogether more ragged than allowed in conventional econometric models.
This volume, retailed as a collection of previously published papers over the past four decades but actually more like a running commentary with selective revisions and new additions, suggests that Mandelbrot himself has moved closer to the econophysicists, perhaps due to his own success in convincing the physicists and relative failure in connecting with economists. Because of this shift, I doubt that any financial economist picking up this book would readily grasp the tenor of Mandelbrot's recent thought without a prior introduction to a primer on multifractals, perhaps augmented with his more recent Selecta volume (1999, Multifractals and 1/f Noise, NY: Springer Verlag).
If there has been a common thread throughout Mandelbrot's economics, it is the conviction that "the essential role of a Bourse is to manage the discontinuity that is natural in financial markets" (ibid, p. 68). His attempt to express this geometric insight has assumed two formats over time, both linked but not well integrated with each other, undergoing shifts in emphasis throughout the period represented in this volume. In the first, one approaches time series of prices as an unabashedly stochastic phenomenon and asks for the most cogent and parsimonious interpretation of the evidence. In 1963, Mandelbrot caused a furor by asserting drat the Gaussian model was a poor fit, and that the more general Levystable family of distributions, derived from a more general limit theorem, gave a better characterization. Over time, Mandelbrot has backed away from this claim as it has come under sustained fire from within the economics profession, but also as he came to appreciate that various stochastic characterizations constituted a continuum, with the Gaussian at one extreme, the lognormal an intermediate case, and Levystable distributions as the "wild" other extreme. Given the family resemblances, it was deemed unlikely that the question of stochastic characterization could be presented as a dichotomous 'either/or,' much less distinguish between long dependence and a marginal distribution with infinite variance, and therefore Mandelbrot now has relinquished many of his earlier claims for generality and simplicity. For instance, he no longer champions a fearless indeterminism (p. 16), and indeed, has forsworn the goal of a general stochastic characterization applicable to all markets (p. 13).
In the second format, the geometric characterization of the price series assumes pride of place while probability takes a backseat; and Mandelbrot reminds us that it was his early work on finance that led to his more famous work on fractals, rather than vice versa. Yet sometime in midcareer, Mandelbrot realized that price time series were not strictly selfsimilar, but rather selfaffine (literally, globally nonfractal). This led to his more recent theoretical commitment to stationarity and scaling as the effective equivalents of conservation laws in physics: unshakable theoretical commitments, whether they areempirically true or not, or as he writes, Hamiltonians allow physics to explain scaling. But those laws have no counterpart in finance (p. 113). Some will feel we are left with a radically undermotivated modeling strategy, which consists of producing computer simulations of price time series which mimic the movements of observed prices by means of deterministic iterative algorithms which squeeze, slice, dice and otherwise massage simple splines and, more curiously, the time axis as well. This constitutes the model that Mandelbrot apparently now favors, at least for first differences of corporate share prices, a combination of fractional Brownian motion in multifractal time. Hence we are left with the portrait of someone who rejects the orthodoxy in modern finance because he believes his geometric characterization contradicts lognormality, ARCH models, the Ito calculus, and most of the rest of the accoutrements of financial economics. By all accounts, he no longer grapples with actual empirical price series as he did in the 1960s; in this second phase we are squarely confined to the realm of stylized facts.
In the shift from the first to the second narrative, we seem to have come quite a distance from the Mandelbrot of the 1960s, even though he himself gives little indication of any analytical rupture. The absence of specific model motivation was acceptable in the 1960s, since the point of the crusade then was to insist that the unwavering adherence to the Gaussian distribution was neither so innocent nor harmless as economists (still) appear to think. Mandelbrot's retreat from the Levystable generalization in the interim, however, has left him in much the same uncomfortable position as those whom he criticizes. For instance, his recent model resembles Peter Clark's (1973, "A Subordinated Stochastic Process Model with Finite Variance," Econometrica, 41, pp. 13555) subordinated stochastic process to a much greater degree than he might be willing to admit; and Clark's model was explicitly intended to provide a more neoclassicallyfriendly alternative to Mandelbrot's original assertions. The recourse to multifractals appears to multiply parameters in much the same way that ARCH and other curvefitting techniques do; and hence it becomes much harder to distinguish which simulation exercise should be deemed as coming off 'better' in the race to mimic price movements, a dire Red Queen tournament that might be considered as having begun with the chartists.
What seems missing from the controversy is adequate consideration accorded to what renders the phenomena specifically economic. For instance, physicists can legitimately entertain the notion that time is a relativistic variable; but why should economists do so? Further, if we must assume for the purposes of statistical estimation that the Brownian motion of prices is statistically independent of the temporal driver, doesn't this conflict with the ingrained notion that price and quantity are interrelated in a specific market? And while there are a surfeit of lapsed physicists drawing their paychecks from brokerage houses, is there something more than this that specifically encourages these particular formalisms in finance, in the absence of any comparative study of the possible selfaffine character of price movements in retail or consumer markets? Mandelbrot's geometric eye should provoke a more serious reconsideration of the aims and goals of quantitative empiricism, something sorely lacking since the old Measurement without Theory controversy.
Mathematical Reviews %Bing Hong Wang (PRCHEFMP; Hefei)
Monatshefte fr Mathematik % (Austria)
% 2000 % P. Schmitt (Wien)
Today, the finance market is a major target for applications of mathematics. In this volume of his Selecta, Benoit Mandelbrot, (also) a pioneer in this field, assembles his papers on this subject (from 1960 onwards) and supplements them with newly written chapters. Thus it is both a monograph of the subject (by one of the masters) and an interesting source of its history.
Nature % Feb 19, 1998, 391 %Ian Stewart
(Math, U. of Warwick, Coventry, UK)
money spinner. In the late 1950s and early 1960s, Benoit Mandelbrot, then a young and relatively unknown researcher at IBMs T. J. Watson Research Center, devoted a lot of his time to problems in economics and finance. His ideas in these areas formed a tiny part of a huge body of work, ranging from rainfall statistics to linguistics, that led him to create the concept of the fractal, for which he is now famous all over the world.
Fractals and Scaling in Finance brings Mandelbrots work full circle, applying todays mature fractal geometry to the problems that plagued him 40 years ago. It is a typically Mandelbrotian mix of reprinted papers, commentary, unpublished results and new work hot off the press. Its focus is simple: what is the structureif anyof financial data? Faced with [the worlds stockmarkets] unpredictability, classical mathematics took one look at the financial world, deemed it to be random, and the paradigm was set. The great triumph of modern financial mathematicsthe famous BlackScholes equation of 1972, without which there would be no derivatives marketis based on a purely stochastic model
The evidence that market data possess hidden structure is becoming overwhelming But what kind of structure do market data possess? According to Mandelbrot, their central feature is scaling. Roughly speaking, the smallscale fluctuations of the market mimic the largescale ones, but on a compressed timescaleThis implies, for example, that jumps in market value can be bigger, and more rapid, than conventional statistics would allowIn a way, the entire book is the story of Mandelbrots intellectual voyage into powerlaw territory, as he refined his early observations into more sophisticated, but always simple and elegant, models of the irregularities of financial time seriesThis is not an easy book to read, but once one gets into the flow of ideas it is rewarding and full of insight, and should be on every chartists bookshelf. There is one weakness, though, that will be apparent to any child of the computer age: a neglect of modern data. Mandelbrot explains that, although extensive data are now available, he is ill equipped for empirical work. Perhapsbut isnt that what postdocs are for?
Neue Zrcher Zeitung %
Feb 25, 1998 % George Szpiro
[Includes 16 old] articles, many of which were breakthroughs...Unfortunately, reading this book is extremely irritating ...It is the psychogram of a conceited genius...Can be understood perhaps if you consider that success has been denied to him for a long time...He failed to receive the highest award in mathematics, the Fields Medal...and had to be satisfied with the Wolf Prize for Physics.
Physics Today % Aug 1998 % Nigel Goldenfeld (Phys, U. of Illinois, UrbanaChampaign)
the stock market: brownian, gaussian or mandelbrotian. Most of us would probably be prepared to agree that stock price changes are not Gaussian; nor are they examples of randomwalk behavior. However, events such as the 1987 crash, World War I and the crash of 1929 are extreme and properly regarded as outliers. What about business as usual?
The answer, of course, depends on whom you ask. On one hand, Burton Malkiels wellregarded semipopular book on finance, A Random Walk Down Wall Street (Norton, 6th edition, 1996), takes its title and its theme from the notion that stock price changes follow a Brownian motion. Virtually every textbook on advanced finance takes the Brownianmotion description as its starting point, and the celebrated BlackScholes formula for option prices is based upon this description.
On the other hand, there is Benoit Mandelbrot. No reader of physics today can be unaware of the enormous impact made by Mandelbrots The Fractal Geometry of Nature (Freeman, 1982), which introduced many to the notions of fractal dimensions, scaling and selfsimilarity and spawned a host of coffeetable imitations.
What is perhaps less well known, however, is that some of Mandelbrots earliest forays into fractals involved a detailed analysis of the time series for cotton prices in New York. His shocking conclusion, published in 1963, was that the time series was in no way Gaussian. In fact, he argued, the departures from normality could be accounted for by using distribution functions with infinite variance, which are termed Lstable. Mandelbrot examined the convergence in sample number of the variance of the logarithm of the daily price changes and found erratic variation rather than convergence.
Subsequently, his student Eugene Fama (who has himself enjoyed a distinguished career in finance) examined the time series for the 30 stocks in the Dow Jones industrial average, finding no exceptions to the longtailed nature of the distributions observed.
The implications of these and subsequent findings are profound, yet it is fair to say that the work was practically ignored by economists and practitioners of finance. Even today, the problem of fat tails is swept under the rug by the vast majority of financial risk managers, even though the phenomenon is sufficiently widespread and well recognized as to have earned its whimsical name.
Mandelbrots heirs are primarily physicists who enter the field of finance, recognize the fundamental importance of fat tails and then elaborate on and extend his suggestive results. This is something of an ironic development, as Mandelbrot takes pains to emphasizeand reflects the close intellectual relationship between finance and physics: Brownian motionwas in fact anticipated by Louis Bachelier five years earlier Elements of Mandelbrots work in the early 1960s, which superseded Bacheliers analysis just as it was becoming widely accepted, arguably anticipate some of the concepts of scaling and renormalization, which were a focal point of physics during the 1970s.
The concepts of fractional Brownian motion and multifractals, which are still frontier topics of research in physics and academic finance (as practiced by physicists), were introduced by Mandelbrot in the late 1960s and 1970s
Fractals and Scaling in Finance is a characteristically idiosyncratic work. At once a compendium of Mandelbrots pioneering work and a sampling of new results, the presentation seemed modeled on the brilliant avantgarde film Last Year in Marienbad, in which the usual flow of time is suspended, and the plot is gradually revealed by numerous by slightly different repetitions of a few underlying events.
As Mandelbrot himself admits in the preface, the presentation allows the reader unusual freedom of choice in the order in which the book is read. In fact, I enjoyed this work most when I read it in random order, juxtaposing viewpoints and analyses separated in time by three decades and making clear the progression of ideas that Mandelbrot has generated. These include the classification of different forms of randomness, their manifestation in terms of distribution theory, their ability to be represented compactly, the notion of trading time, the importance of discontinuities, the relationship between financial time series and turbulent time series, the pathologies of commonly abused distributions (particularly the lognormal) and a catalog of the methods used to derive scaling distributions, both honest and fallacious.
Mandelbrot writes with economy and felicity, and he intersperses the more mathematical sections with frank historical anecdotes, such as the events that led up to his work on cotton pricing and the embarrassment caused by interpreting US Department of Agriculture data for weekly averages as Sunday closing prices. There are many fascinating asides on a variety of topics, ranging from the importance of computer graphics in science to the distribution of insurance claims resulting from fire damage. In some places, the format of reprinted (but slightly edited) versions of classic papers allows Mandelbrot the surreal luxury of reviewing not only the content, but also the style and presentation of his work. And if all this were not enough, there are guest contributions from Eugene Fama, Paul Cootner and others.
This volume is not intended to be a textbook of modern finance, and it will probably infuriate those seeking a balanced and systematic expositionbut to criticize the volume on that account would be churlish. The reader who is openminded and prepared to indulge one of our more influential and original thinkers will be amply rewarded.
All in all, this is a strange but wonderful book. It will not suit everyone s taste but will almost surely teach every reader something new. What more can one ask?
Statistica Applicata % 10, 1988.3, p 504
Benoit B. Mandelbrot achieved great fame in the scientific world after the publication of "The Fractal Geometry of Nature," which was followed by a variety of books and papers on fractals, a topic that he introduced. Probably this great mathematician is not equally known except among specialists for his innovative studies regarding financial phenomena (in particular variations of financial prices and the distribution of personal income), starting with celebrated articles in the early '60s. Reprints of Mandelbrot's papers on the subject constitute about half of this volume, while the other half for the most part non mathematical has been written by the author expressly for the publication of this book. Mandelbrot's approach to finance stems from a critique (always sustained and actually strengthened even more) of the traditional use of random walks, of Brownian motion, and of martingales to describe and, for some, also to explain price variation or financial markets. His dissatisfaction regarding the use of Brownian motion is based on a detailed analysis of a long series of prices. From these he derives innovative proposals and much more general schemes, which, however, preserve the fundamental stationarity and scaling properties typical of Brownian motion. Mandelbrot's proposal is to allow us to abandon the Gaussianity of Brownian motion, which can describe only mild variations found in many physical phenomena, and to tackle wild variation, which can exhibit genuine discontinuities and turbulences peaked around particular periods.
Statistical Papers % 41, (2), p.245246, 2000 % Christian Kleiber (Dortmund, DE)
In the last forty years, Benoit Mandelbrot has made his mark in an impressive number of scientific disciplines. To a wider audience he is probably best known for his work on fractal geometry, but he has also contributed substantially to economics and physics, among other fields. The book under review is the first of a multivolume series of Mandelbrots Selecta, dealing with his work in economics. Future volumes will be devoted to hydrology, turbulence, and other physical phenomena. The general concept is to present some of Mandelbrots classic papers in a field, along with unpublished work. For the present volume, this means that about half of the material has not been available before. The bestknown papers of those included are probably The variation of certain speculative prices, originally published in the Journal of Business in 1963, and The ParetoLvy law and the distribution of income, originally published in the International Economic Review in 1960. Seven further articles are reprinted here. In addition, one gets a number of expository papers, comments (sometimes on the work of others) and unpublished working papers, mostly from the 1960s. Eugene Famas wellknown article Mandelbrot and the stable Paretian hypothesis (Journal of Business, 1963) is included as a guest contribution. Most papers are presented in their original form, however, the titles and the terminology have sometimes been changed. For example, what Mandelbrot used to call a ParetoLvy or stable Paretian distribution in the 1960s is now called an Lstable distribution, and what used to be the strong Pareto law is now the uniform Pareto law.
The general theme is selfaffinity (or powerlaw behavior, or scaling) in economics. For example, in empirical finance Mandelbrots contributions are twofold: first, he observed that quite a few return distributions were incompatible with Gaussianity and suggested, in order to be able to draw on the invariance principles of probability, to replace the normality assumption with that of a nonnormal (infinite variance) stable distribution. Second, it turned out that martingale models of the prices themselves could be replaced, at least on a deformed time scale (trading time), by something more dependent, a fractional Brownian motion. The first idea, in Mandelbrots words: taildriven variation, leads to hyperbolically decreasing tails, the second, dependencedriven variation, to hyperbolically decreasing autocorrelations of the increments of such processes. It took some time to nest these two features within a more general model, the multifractal model of asset returns, which has only recently been worked out in detail and is still unpublished. Chapter 6 of this book nevertheless provides the basic ideas.
As Mandelbrot states in the preface, his methods of investigation are those of practicing theoretical and computational physicist. Hence, the presentation is frequently not mathematically rigorous  this is, however, amply compensated by a wealth of ideas. For example, it is impressive to see how many results in e.g. Samorodnitsky and Taqqus (1994) Stable NonGaussian Random Processes have been inspired by Mandelbrots work of the 1960s and 1970s.
In spite of the title, this book is not exclusively concerned with financial economics. It also contains some work on the distribution of income and the size distribution of firms, Fractals and Scaling in Economics would be a better title. In view of the recent boom in empirical finance, the publisher apparently hopes to boost sales this way.
Of the newly added material, I particularly enjoyed chapter 4, Sources of inspiration and historical background, where Mandelbrot finds scaling, apart from economics and physics, in the laws of allometry, in the work of Jonathan Swift, or in a line of William Blakes. Also, his comments on whole branches of the literature are always entertaining: the most popular brand of nonlinear models for stock returns, for example, is dismissed on p.44 as patchworks of quick fixes called ARCH models.
To sum up, this is a most useful collection of Mandelbrots work in economics, it provides an excellent starting point for anybody interested in the origin of many current topics in empirical finance or the distribution of income.
Wirtschaftsinformatik (BRD) %
41, 1999.1, p 95
FRACTALES, HASARD ET FINANCE
AFP: L Information la Source
Peu de personnes savent que la gomtrie fractale est ne des travaux que Benot Mandelbrot, son inventeur, avait consacr la finance dans les annes soixante. Prs de quarante ans plus tard, le mathmaticien convie le lecteur utiliser ces mmes concepts dimprvisibilit pour dcrypter le comportement des cours de la Bourse et tenter une valuation raliste des risques financiers. Tout avait commenc pour Mandelbrot, en 1960, [quand] il dcouvrit une symtrie entre les grandes et les petites chelles: la courbe dvolution sur une semaine tait en effet semblable celle dune dizaine dannes. Ce sera le dbut de lpope de ces objets rugueux, poreux ou fragments et invariants toutes les chelles, nomms objets fractals. Cet insatiable curieux a depuis appliqu sa vision [quantit dautres domaines].
Capital % Jan 1998 % JeanFranois Rouge
la mthode d un matheux clbre pour gagner en bourse. Le professeur Ian Malcolm, le hros de Jurassic Park et du Monde perdu & emprunte la plupart de ses ides un mathmaticien franais de 73 ans, Benot Mandelbrot& Malcolm invoque en effet une thorie mathmatique, dite du chaos... On parle aussi de gomtrie fractale. Le pre de cette science en vogue (et donc souvent trahie) vit New York et enseigne luniversit de Yale.
Jamais, depuis Einstein, un thoricien dont les crits sont pourtant de lhbreu pour la plupart des mortels, navait bnfici dun tel effet de mode, y compris en librairie
[Dans son nouveau livre], Mandelbrot renoue avec un thme de recherche quil avait abord ses dbuts, dans les annes 60: la prvision et, surtout, la prvention des chocs boursiers. Les fractales, qui permettent par ailleurs de mesurer prcisment la longueur dune cte accidente ou de prvoir la croissance de la vgtation dans une fort, peuvent en effet dterminer la volatilit dune action ou dun indice. Une connaissance qui permet ensuite de fixer les rserves ncessaires une banque ou une compagnie dassurances pour sen tirer au mieux en cas de tempte boursire ou montaire.
Le Figaro % 13 Dc 1997 % Georges Suffert
le dterminisme invisible des cours de la Bourse. les surprenantes intuitions de Benot Mandelbrot: L inventeur des fractales s attaque aux lois de la finance. Ou pourquoi l ordre cach du hasard renvoie dos dos pessimistes et optimistes.
Benot Mandelbrot est un personnage qui chappe toute classification. A premire vue, on pourrait le prendre pour lhritier conjoint dEinstein et du professeur Nimbus. Il donne limpression de se mouvoir dans un univers sans grand rapport avec le ntre; il voit, il pressent, il formule dtranges rapports mobiles entre des formes apparemment figes jamais. Depuis des annes, il dtecte le dsordre invisible qui se dissimule la surface des objets, dans des structures de lignes apparemment droites. Il est lun des fils de lalatoire, il poursuit des ruptures que nous ne percevons mme pas. Et dun mme mouvement, il reconstitue une forme de rgularit, dapparent dterminisme derrire le fourmillement des apparences dsordonnes. Les fractales furent la dcouverte qui obligea ses collgues mathmaticiens, gologues et physiciens admettre que Mandelbrot avait de surprenantes intuitions. On se souvient de limage dsormais classique qui popularisa ces fameuses fractales: il suffit dobserver sur une carte la cte de la Normandie Aujourdhui, Benot Mandelbrot, polytechnicien et normalien, est reconnu, aux EtatsUnis du moins, comme un exceptionnel observateur du rel. Il commence par voir, puis il fait intervenir les mathmatiques. Si lon veut mesurer limportance de son apport, il suffit douvrir les ouvrages actuels sur le chaos ou les mlanges (chez Odile Jacob): le systme des fractales soustend lessentiel des descriptions prsentes.
Mais aujourdhui Mandelbrot se lance dans une chasse particulirement fascinante: les cours de Bourse obissentils des lois connues?
Lalatoire ordinaire: tout se passe comme sil y avait du dsordre dans les variations des Bourses ce problme fascine, sembletil, Mandelbrot. Il a commenc par sinterroger sur le modle propos par Louis Bachelier en 1900. Les prix, daprs ce chercheur lointain, se promnent dans lalatoire ordinaire. Do un mouvement brownien classique. En 1963, puis en 1965, Mandelbrot propose ce quil tiquette sous le terme de brownien fractionnaire.
Hasard sauvage: cette date, notre chercheur a repr sur courbes et quations deux types distincts de hasard sauvage: les vnements catastrophiques isols (effet No) et les alternances rgulires de vaches grasses ou maigres (effet Joseph). On aboutit des courbes imprvues, trs diffrentes de celles de Bachelier et de ses successeurs
On fera remarquer que pour le moment, la thorie de Mandelbrot ne sert rien: pas question de gagner en Bourse sans risque derreurs. Mais lintrt rel nest pas l: ce que notre chercheur commence distinguer, ce sont ces lois invisibles et inconnues qui gouvernent ce que nous appelons hasard; encore une fois, une espce de dterminisme invisible qui surgirait de lalatoire luimme. Il y aurait un ordre trange dans limmense dsordre de lunivers apparent.
Un petit ouvrage lire pour ceux qui ne reculent pas devant les courbes et quelques quations. Mais on peut sauter ces passages.
Kwantitatiere Methoden (NL) % E. Omey
Le Lgitimiste % Novembre 1997
La collection Champs chez Flammarion, trs bon ensemble de textes de haute vulgarisation scientifique, vient de faire paratre deux livres qui valent la peine d tre lus, mme s ils ne sont pas, il s en faut, des reflets de nos ides ou de nos proccupations [Le deuxime de ces] ouvrages est sign Mandelbrot, mathmaticien connu pour ses travaux sur les fractales (objets mathmatiques lis la thorie en vogue du chaos): Fractales, Hasard et Finance. Par son caractre (relativement) appliqu aux questions conomiques, ce livre est (toujours relativement) beaucoup plus abordable que bien dautres et constitue ainsi une bonne entre dans une thorie qui marque considrablement notre temps. La lecture ncessite cependant un bagage scientifique solide, particulirement en probabilits, discipline lie, comme chacun sait au hasard.
Libration Eureka %Mardi 24 Fvrier 1998 % Sylvestre Huet
En appliquant sa thorie des fractales, les chiffres qui servent crire le chaos, la finance, Benot Mandelbrot avertit que le risque boursier est plus grand que ne le disent les courtiers.
% Votre livre distingue les hasards bnins et sauvages . Que signifientils?
M. En science, le hasard n est pas le sort personnifi, mais uniquement une mesure de notre ignorance. Dans certains domaines comme la physique classique, cette ignorance est contrlable par les mathmatiques. Cest un hasard que jai baptis bnin. Si lon attend suffisamment longtemps, on dcouvre lordre cach. Si vous jouez pile ou face vous aurez une statistique trs simple de 5050. Si vous coutez une radio sans viser un metteur vous entendez un bruit de fond. Il provient du hasard, mais on ne peut lliminer car il bouge sans arrt autour dune intensit bien fixe.
Mais il y a un autre hasard, le sauvage. Il est trs vilain, car il ne permet pas de raisonner en termes de moyennes. Si vous prenez dix villes de France au hasard et si vous ratez Paris, Lyon et Marseille, vous allez faire chuter la taille moyenne dans votre chantillon. Si vous prenez dix villes dont Paris et neuf villages, la moyenne nautorise aucune conclusion sur les populations de villes tires au hasard. Un autre exemple spectaculaire, cest la distribution des galaxies dans lUnivers. Classiquement, les astronomes partaient de l ide que lUnivers, trs grande chelle, prsentait une distribution uniforme des galaxies. Donc que la notion de densit moyenne de matire avait un sens. Or, plus on voit loin, et plus on distingue dnormes vides, qui dmolissent cette ide. En fait, la distribution des galaxies semble tre un exemple merveilleux de hasard sauvage. Les techniques usuelles ne permettent de tirer aucune conclusion sre. Il faut donc changer de mthode mathmatique.
% C est l ce que vous proposez d utiliser les mathmatiques fractales. Que sontelles?
M. Une fractale est un objet mathmatique que l on peut couper en petits bouts et dont chaque bout prsente la mme structure que le tout. Le choufleur est une trs jolie fractale naturelle. Chaque morceau que vous dtachez prsente la mme structure que le tout, et ainsi de suite. Avec deux bornes de taille, suprieure et infrieure, videmment, alors que son analogue mathmatique peut tre sans limites. Lhomme aime bien cette vision hirarchique, lembotement des structures. Le point central de ma dcouverte, cest quil ne sagit pas seulement dune astuce mathmatique mais que cest une proprit fondamentale de trs nombreux objets naturels. Quand on combine hasard sauvage et fractalit, on a dun ct une mauvaise nouvelle: difficult accrue de la comprhension et de la prvision. Et, de lautre, une bonne nouvelle: si lobjet, ou la dynamique, peut tre dcrit laide dun nombre fractal, on se retrouve avec un objet mathmatique relativement simple, puisquil est fond sur une invariance, le concept de base de la science. Dans une fractale, il sagit dinvariance dchelle: les proprits sont les mmes, quelle que soit lchelle laquelle on les regarde.
% Avec cela, vous pouvez aider les financiers grer le hasard sauvage de la Bourse?
M. Prvoir les cours de la Bourse avec les mathmatiques n est pas un espoir srieux. Si une telle formule existait, son objet disparatrait d ailleurs automatiquement, car tous les acteurs boursiers auraient la mme information sur lvolution des actions, ce qui changerait leurs dcisions. Les grands changements de prix sont, pour presque tout le monde, imprvisibles. Mais, si on pouvait mieux en valuer statistiquement les risques, on pourrait les amortir. Or, les fractales permettent justement dtudier simplement les cours, le comportement dune Bourse, en indiquant son degr de variabilit par un seul chiffre, la dimension fractale des courbes de prix des actions. Un chiffre est compris entre 1 (la ligne droite) et 2 (la surface), qui peut aider mieux apprcier le risque de chute brutale. Mon travail ne promet pas de bnfice pour un particulier, mais il peut aider mieux chiffrer les rserves obligatoires des banques, assurer des investissements de fonds de pension ou rguler un peu le march, le protger des soubresauts et des faillites retentissantes. On veut, en somme, moyenner les risques pour avoir le moins de fluctuations possible. Mais, pour faire cela, il faut estimer les risques de manire correcte. Or, lexprience le prouve, les risques sont beaucoup plus grands que ne le disaient les thories conomiques. Regardez le nombre de faillites totales de grands portefeuilles, tenus pourtant par des experts.
% Pouvezvous prvoir les bulles boursires, ces hausses des cours qui semblent dconnectes de l conomie relle, comme celles qui viennent d imploser en Asie?
M. En 1966, j ai dcrit dans un article une Bourse trs rationnelle et pourtant telle que les prix paraissent monter sans arrt, toujours rationnellement, puisque chaque moment, la persistance de la monte est plus probable que son interruption. Mais, corrlativement, la valeur de la chute possible monte elle aussi. Et, plus on attend, plus la chute est rude. Donc le risque augmente constamment. Cela tait contenu dans les formules mathmatiques produisant ces fameux courbes fractales qui ressemblaient tant aux courbes relles. Lors du krach du 19 octobre 1987, des financiers mont dit: Cest exactement le comportement que tu dcrivais en 1966. Anticiper la prsence de grosses bulles, montrer quelles peuvent tre parfaitement rationnelles est un triomphe de la thorie. Dautant plus, finalement, que lon peut, avec les mathmatiques fractales, les dcrire et prvoir pour elles un risque plus lev que ne le pensaient les thoriciens de la finance sans pour autant expliquer les raisons conomiques de ce que jobserve, ce qui nest pas de mon ressort.
Pour la Science % Janvier 1998 %Ivar Ekeland (U. ParisDauphine)
Benot Mandelbrot & n est gure modeste, et a d excellentes raisons de ne pas l tre& Le but avou de son dernier livre est de montrer l unit de sa pense scientifique et de raconter comment il a eu raison avant tout le monde
Les ides qui animent jusqu aujourdhui loeuvre de B.Mandelbrot et les centres dintrt auxquels il na pas cess de sattacher pendant un demisicle [sont] la rpartition des revenus, lvolution des cours boursiers, les crues de Nil et la frquence des mots B.Mandelbrot insiste plusieurs reprises sur le fait quil avait peu prs tout dit ds le dbut.
Ses ides sont simples et robustes Tirons au sort une suite de N valeurs X1, X2, X3, indpendantes et quirparties dans lensembles des nombres rels. Trois types de situations sont possibles. Premirement, le cas classique, o les moyennes X1, (X1 + X2)/2 , ( X1 + X2 + X3)/3 convergent rapidement vers une valeur finie et certaine (loi des grands nombres), et o lcart la moyenne, pondr convenablement, converge vers une variable alatoire dont la distribution est reprsente par la trs fameuse courbe en cloche de Gauss (thorme central limite).
Deuximement il y a le cas lent, reprsent par exemple par la loi lognormale.
Troisimement le cas sauvage, reprsent par la loi de Cauchy, dont lesprance et la variance sont infinies; dans ce cas, les moyennes ont exactement la mme distribution que chacun des tirages individuels. aussi loin quon aille, on ne vera jamais sinstaurer de compensation entre les diffrents tirage: la moyenne est aussi incertaine que chacune des preuves.
Ce sont des cas extrmes, et il y a entre eux tout un continuum de cas intermdiaires [relatifs ] ce quil appelle joliment l effet No et l effet Joseph.
Ces lois ont dautres proprits intressantes, mais leur intrt principal est quelles fournissent des modles qui paraissent mieux adapts que la simple loi normale pour reprsenter certains phnomnes, comme, par exemple, les crues du Nil ou les cours de la Bourse. On ne peut qutre frapp, une fois de plus, par le caractre purement visuel des analyses de B.Mandelbrot. Comme il le dit luimme: Prcisons le rle des images. Jinsiste sur le fait que la capacit dimiter est dj une forme de comprhension. Mais jai toujours dit, et je reconnais volontiers, que les images doivent ncessairement tre suivies de commentaires statistiques objectifs.
Ce nest pas ce que la plupart dentre nous entendent par comprhension. [La] recherche des causes est dautant plus importante quon ne sintresse pas ces trajectoires pour le plaisir, mais pour raliser certaines oprations de couverture
Quiconque veut proposer un nouveau modle doit se soucier de savoir ce que deviennent [les] stratgies de [BlackSholes]. Ce n est pas le souci de B.Mandelbrot: il lui suffit d avoir vu, et de pouvoir montrer. B.Mandelbrot est un mathmaticien pur.
La Recherche % Francis Wasserman
L inventeur des fractales propose ici une & recherche qui se focalise sur les invariances dchelle, la base des structures fractales dont il expose ici les principales notions.
Lauteur est amen dfinir trois tats du hasard: bnin (celui de la statistique classique et des lois de Gauss), lent ou sauvage (ceux de la finance) Le rle et lusage du hasard sont ainsi la charnire dune rflexion qui articule un dialogue constant entre lconomie et la physique, entre alatoire et non alatoire cheminement fcond pour les analogies et les rapprochements quil a suggres (crues et mouvements boursiers par exemple).
Louvrage prsente bien les fondements statistiques des fractales trop souvent masqus par la vulgarisation. A ce titre, la lecture de cette synthse sadresse plutt un public averti.
La Tribune %
2 Fvrier 1998 % Lysiane J. Baudu
le choufleur et les marchs. Le mathmaticien Benot Mandelbrot, presque autodidacte, est le pre de la gomtrie fractale. En l appliquant aux marchs financiers, il fait comprendre leur volution erratique.
Benot Mandelbrot [est] clbre dans le monde des mathmaticiens pour ses thories indites sur la gomtrie fractale, qui fait comprendre lirrgularit des formes, comme celle des flocons de neige ou des rseaux de rivires. Des thories quil cherche mettre au service de la finance, depuis quelques annes, pour comprendre lvolution erratique des cours de la Bourse. Mais pour expliquer ce concept, il saide dun choufleur et explique que ce lgume [a une] surface ruguese. Alors que loue, le chaud, le got, ont t trs bien tudis par la science, le sens du rugueux est rest quasiment inexplor, remarquetil. Et il ny a pas que le choufleur dont la surface est rugueuse, les prix des actifs financiers aussi, Cette montagne de science, du haut de son 1,90 mtre, enfourche son cheval de bataille. Pour expliquer que lvolution des cours des actifs est discontinue, que les prix peuvent dcaler brutalement, et non pas de faon graduelle et continue. Comme le beau temps et le mauvais temps en somme Il ne sagit en aucun cas de prvoir un krach, lhomme est trop raliste pour cela. Il essaie simplement de mesurer la rugosit des cours, et den ramasser la complexit en une formule simple...
Les thories de Benot Mandelbrot sont clbres dans le monde entier. De grandes banques amricaines avaient dbloqu un gros budget, une quipe de plusieurs dizaines de personnes pour travailler sur le chaos, raconte un professionnel, maintenant ce sont effectivement les thories fractales qui sont la mode. Mme si certains ne croient pas franchement au bienfond de la thorie fractale appliqu la finance, ils restent impressionns par les recherches et le savoir du mathmaticien de Yale.
MULTIFRACTALS AND 1/f NOISE
L Enseignement mathmatique %
45 (3/4), 1999.
Mathematical Reviews %Daniel J. M. Schertzer & Shaun M. Lovejoy
The book primarily emphasizes the importance of two papers: J. Berger and B. B. Mandelbrot (1963), and B. B. Mandelbrot (1974). The former is& presented as a main historical step to understanding clustering/intermittency with the help of strongly nonGaussian noises. This indeed corresponds to an important step towards the use of fractal dimension to characterize intermittency.
The 1974 paper deals with a broad and significant generalization of the pioneering multiplicative cascade model of A. M. Yaglom (1966) which was built up in order to understand turbulent intermittency. Indeed, it is physically argued that the "microcanonical" constraint (i.e. strict conservation of the energy flux) should be replaced by a less demanding "canonical" one (i.e. conservation of the energy flux under statistical average). The latter yields a much more variable energy flux.
That paper is followed by a revised English translation of a more mathematically oriented twopart companion paper [B. Mandelbrot (1974)] that presents several conjectures. The next chapter a "guest contribution" corresponds to an English translation of J.P. Kahane and J. Peyrire, Advances in Math. 22 (1976), presents exact mathematical results on multiplicative cascade processes.
There is a sharp difference that one needs to underline between [J. Berger and B. B. Mandelbrot, op. cit.] and [B. B. Mandelbrot, op. cit.], and the corresponding parts of the book: the former deals with additive processes [Comment by BBM: Not in the least], the latter with multiplicative ones.
Zentralblatt der Mathematik %
Yimin Xiao
For selfsimilar fractals, the most important aspect is the measure of roughness or irregularity called fractal dimension. Selfaffine fractals intrinsically are far more complicated in many ways. In this book, Mandelbrot intends to show that selfaffinity helps apprehend and organize the baroque wealth of structure found in nature. This book is mainly devoted to two broad topics: 1/f noises and multifractals, which arose independently and developed as two separate subjects and their histories carry very distinct flavors. Indeed, 1/f noises are of primary concern in signal processing and applied physics; while multifractals began in the study of turbulence and dynamical systems, and also in finance as shown in M1997E. Nevertheless, Mandelbrot defends successfully the view that multifractals and 1/f noises are actually intimately interrelated  they belong to the broad and unified mathematical notion of selfaffine fractal variation which ordinarily implies uniform global statistical dependence, and they are best viewed as being distinct aspects of a wild rather than mild random or nonrandom phenomenon. An 1/f noise is defined as having a spectral density proportional to f B, where B is a constant which is called the spectral exponent. Since 1/f noises with the same spectral exponent can take any of many different forms, understanding such a noise demands more than just the spectrum. The link of 1/f noises to selfaffinity makes it possible to study 1/f noises by introducing a formalism made of scaling laws, renormalization and fixed points. By focusing on three forms of 1/f noise: fractal dustborne noises, multifractal noises, and Gaussian 1/f noises Mandelbrot sketches a broad analytic method for discriminating between those various possibilities. The excellent term multifractal was chosen by Frisch and Parisi in 1985. But constructive and rigorous approach to multifractals as physical models was developed in 1970s by Mandelbrot (see comments about Part IV below). The multifractal formalism centers on two mutually related functions (q) and f() (via the Legendre transform), each can be introduced in several closely related, but nonequivalent ways. Mandelbrot's original approach was based on Cramr's theory of large deviations.
This book consists of four parts. Part I provides newly written introductions to the book which relate diverse models and themes to one another. In particular, Chapter N1 gives a panorama of gridbound recursive selfaffine constructions that allow for a great diversity of behavior, including simplified cartoons for Wiener Brownian motion, fractional Brownian motion, Lvy flight, 1/f noises of several distinct kinds and multinomial multifractals, with a view towards further research topics. Chapter N2 describes how the reprints classified into Parts II, III, and IV fit together historically and examines conceptual connections and relevant historical events. Part II is primarily concerned with selfsimilar unifractal models in nature. The underlying constructions are random Cantor sets and Lvy dusts. Chapter N6 reproduces the important paper of Berger and Mandelbrot (1963) which proposed a new mathematical model based on a Cantor dust to describe the occurrence of errors in data transmission on telephone lines and was the first to interpret the previously esoteric notion of fractal dimension as a fundamental physical quantity. This model is improved and generalized in Chapter N7, in which the concept of conditional stationarity is introduced. Part III concerns the fluctuations called 1/f noises that are dustborne, that is, vary when time belongs to a Cantor or Lvy dust. Such noises are called sporadic or absolutely intermittent. It argues that the WienerKhintchine spectral theory and even the conventional theory of stochastic processes are not enough in investigating such wild noises and begins constructing two generalizations: conditional spectral analysis and sporadic processes. Part IV addresses multifractal measures and turbulence. The centerpiece reproduces [J. Fluid Mech. 62, 331358 (1974)]. Its objective was to show that Kolmogorov's third hypothesis  the probability distribution of the average of the dissipation in intermittent turbulence is lognormal  was untenable. As it turned out that it is of broader impact: it was the first paper to investigate the concept of random multifractal measure. The random fractals in Chapter N15 are created by multiplicative random cascades. Mandelbrot's approach relies upon Cramr's large deviation theory, while in some later investigations, including [Halsey et al], the approach is exclusively analytic. This part also contains the important paper of J.P. Kahane and J. Peyrire [Adv. Math. 22,131145 (1976)], which solves several conjectures about the Mandelbrot's canonical model for turbulence and later has inspired a lot of further research in mathematics.
This volume of Mandelbrot's Selecta is a major contribution to the understanding of wild selfaffine variability and randomness. It is also a mine of other new ideas which are of use to diverse scientific communities from physics, pure mathematics, to finance. Finally, for more topics on multifractals such as multifractal functions, fractional Brownian motion of multifractal intrinsic time and their relevance to finance.
GAUSSIAN SELFAFFINITY AND FRACTALS
Mathematical Reviews %
Michle MastrangeloDehen
This book is a complete and encyclopaedic synthesis of problems and themes in selfaffinity fractals, multifractal geometry and globality. In the early 1960s, Mandelbrot began to explore these subjects; the aim of this book is to contribute to the scope of knowledge in and the development of the field. It addresses numerous old and new problems arising in many varied disciplines: mathematics, physics, engineering, hydrology, climatology, statistics, economics, finance, etc. The first third of the volume consists of extensive introductory material written especially for this book. At the beginning, there is an overview of recent work in fractals and multifractals.
At the end, a very exhaustive and complete bibliography on the different connected subjects (with more than 500 references) may be very useful for researchers or users.
Zentralblatt der Mathematik %
Yimin Xiao
This book contains the author's works in areas ranging from statistics, mathematics, physics, hydrology to economics and finance that first appeared from 1965 to 1988, as well as more than 200 pages of new material written especially for this volume. Most of this book deals with random functions that vary in continuous time and take continuously distributed values. In many cases, they follow the Gaussian distribution. The most important random function in this book is the fractional Brownian motion. A large portion of the book is related to the geometric studies of fractional Brownian motion.
Mandelbrot chooses the letter H to denote this volume among other Selecta because, as he explains, The selfaffinity exponent entered science through my response to remarkable findings of the hydrologist Harold Edwin Hurst. The letter H also refers to Hlder exponent, a mathematical term characterizes the local behavior of a function. Volume H has many close connections to the topics of Selecta E and N. Some of the links are described in Chapters H1 and H30. Compared to these previous volumes, Volume H has a higher level of mathematics and contains many theorems, problems and conjectures phrased in mathematical terms. Together with Selecta E and N, this volume further develops many aspects of the author's famous Essay [The Fractal Geometry of Nature, Freeman (1982)]. Volume H begins with Chapter HO which is an overview of fractals and multifractals. It tackles broad issues and answers diverse questions. Mandelbrot shows that fractal geometry has one focus in mathematics and another in the broadly based discovery that scaleinvariant roughness is ubiquitous (both in Nature and in manmade structures) but can be handled quantitatively.
Fractal geometry is a study of scaleinvariant roughness, and it is a quantitative and organized new language of shape, a toolbox of statistics and data analysis, geared toward the study of wild randomness and variability. Together with Chapter H8, this introduction describes the farreaching and highly relevant historic roots of fractal geometry, the author's principle contributions to various scientific areas related to this book, the content of fractal geometry in 2001 and his view towards further research topics. The rest of this book consists of seven parts. Parts I and II consist of specially written introductions to the book which relate diverse models and themes to one another. The unifying concept is selfaffinity, which is surprisingly rich and involved and is still under development.
Chapter H1 approaches the concept of selfaffinity by concentrating on a quite narrow family of selfaffine curves constructed recursively from an initiator and a generator. It is shown [see also the long foreword of Chapter H24] that small changes in the generator can have spectacular affects on the resulting selfaffine curve. Chapter H2 discusses several different notions of selfaffinity. As a particularly important example of selfaffine Gaussian fractals, Wiener Brownian motion (WBM) is treated in Chapter H3, in which its structure and fractal dimensions are investigated. Chapter H3 not only discusses the classical results on the HausdorffBesicovitch dimensions of the trails and records of WBM, but also describes several old and new conjectures on the HausdorffBesicovitch dimension of the hulls of the Brownian and percolation clusters. Exciting new developments include the recent mathematical proof of Lawler, Schramm & Werner (2000) on Mandelbrot's famous 4/3conjecture about the Hausdorff dimension of the Brownian cluster [M1982FGN, Plate 243], and the proof of Smirnoff (2001) about the corresponding 7/4 and 4/3 conjectures on percolation clusters. These results show that, contrary to the impression that WBM is a wellunderstood, mature topic, the fractaldimensional properties of WBM prove to be multiform, complex and subtle in many ways.
Chapter H4 discusses the Weierstrass family of functions. Even though it is only partly concerned with Gaussian fractals, its topic is highly relevant to this book's goals. The original Weierstrass functions serve as examples of functions that are continuous but not differentiable. These old functions are modified and extended (e.g., adding subharmonics and allowing random phases) to get new Weierstrass functions with increasingly rich invariance. They are useful in diverse ways. Chapter H5 proposes a classification of diffusion processes into isodiffusive and heterodiffusive processes based on their R/S local (also called shortterm) and global (also called longterm) dependence. The existence of global dependence is due to either of two causes, alone or together: global dependence which is labelled the Joseph effect, and fat (heavy) tailedness, which is labelled the Noah effect.
H6 discusses transformation between selfaffine functions and stationary processes, using a logarithmic time clock. In the Wiener Brownian motion case, this time change gives the OrnsteinUhlenbeck process.
Chapter H7 is on empirical powerlaw behavior, 1/f noise and their connections to selfaffinity. This chapter is closely related to Chapter N3 of Selecta N (1999). Chapter H8 tackles topics close to those of the overview chapter HO with more descriptions of the author's personal involvement. It contains a littleknown historical episode concerning random walk, Brownian motion and the role of the eye in assisting scientific understanding and a summary of the author's principle contributions to the topics of this book. H8 also brings together (in chronological order) some personal recollections of improbable encounters with fields, collaborators, referees and editors.
Parts III to VII of this volume consist of reprints of the papers of the author and his coworkers on the topics between 1965 and 1988, with added forewords, annotations and historical remarks. Part III consists of Chapters 9 & 10 and can be regarded as an easier introduction to the subject of this book. Chapter H9 is the oldest paper reprinted in this volume. It links Hurst's power law with selfaffinity, and introduces fractional Brownian motion (without the name). H10 proposes a family of statistical models of hydrology which account for the Noah and Joseph effects in studies of precipitation.
Parts IV and V concern fractional Brownian motion, fractional Brownian surfaces and their usefulness in statistical modelling. Chapter H11 reprints the famous paper of the author with Van Ness (1968) in which they defined fractional Brownian motion and investigated its moving average representation, selfaffinity and long range dependence. Chapters H12 and H13 test the quality of fractional noise as a model of reality and to estimate its parameters. Two approaches are the R/S analysis and spectral analysis, of which the former is discussed in more detail in Part VII. Chapter H14 is complementary to H11 [M & Van Ness (1968)]. H15 describes a fast fractional Gaussian noise generator. Chapters H1720 of Part IV deal with fractional Brownian surfaces [multidimensionaltime extensions of fractional Brownian motion], their mathematical constructions, selfaffinity and applications in modelling turbulence and the Earth's relief.
Part VI intermingles the themes of selfaffinity, various fractal dimensions and multifractals. The results on fractal dimensional properties of selfaffine sets show that, in contrast with the case of a selfsimilar sets where there is a unique fractal dimension, several distinct dimensions are needed for characterizing the structures of selfaffine fractals. Mandelbrot believes that the richness and complexity of the study on selfaffine sets are not purelymathematical but reflect the richness and complexity of nature.
Chapter H21 serves as an introduction on the main points of Part VI. H22 discusses the local and global mass/box dimensions and the gap dimension of selfaffine fractals. H23 deals with selfaffine fractal curves and shows that, walking a divider along a curve yields a local and a global values that may be different from the mass/box dimension of the curve. H24 discusses a result of C. McMullen [Nagoya Math. J. 96, 19 (1984)] which shows that the HausdorffBesicovitch dimension of a recursively constructed selfaffine fractal may be strictly smaller than its local box dimension, and the implications of this result to new developments in fractal geometry. One of the implications is, as Mandelbrot suggests, the special standing of the HausdorffBesicovitch dimension in fractal geometry may have lost.
Part VII studies R/S analysis systematically. It contains reprints of the two papers with J. R. Wallis [H25 & 27] which discuss the foundation of the R/S method in depth and address issues in hydrology and geophysics, a mathematical paper of himself [H26] on limit theorems for the selfnormalized bridge range, and papers on applications of R/S analysis to studying the secular motion [H28], linguistics [H29], economics and finance [H30].
To summarize, Volume H of Mandelbrot's Selecta is a major contribution to the understanding of wild selfaffine variability and randomness. I believe that, similar to M1982FGN, it will have a great impact on future research in mathematics, statistics, physics and other applied areas.
FRACTALS AND CHAOS
Foreword by Peter W. Jones
(Professor of Mathematics, Yale U.)
It is only twentythree years since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot set. The graphics available at that time seem primitive today, and Mandelbrot's working drafts were even harder to interpret. But how that picture has changed our views of the mathematical and physical universe! Fractals, a term coined by Mandelbrot, are now so ubiquitous in the scientific consciousness that it is difficult to remember the psychological shock of their arrival. A twentyfirstcentury researcher does not think twice about using a computer simulation to begin the investigation of a problem; indeed, it is now routine to use a desktop computer to search for new phenomena or seek hints about research problems. In 1980 this was very far from the case.
When a paradigm shift hits, it is rarely the old guard who ushers it in. New methods are required, and accepted orthodoxy is often turned on its head.
Thirty years ago, despite the appearance of an avant garde, there was a general feeling in the mathematics community that one should distrust pictures and any information they might carry. Computer experiments had already appeared in the undergraduate physics curriculum, but were almost nonexistent in mathematics. Perhaps this was due in part to the relatively weak computers then available, but there were other aspects of this attitude. Abstraction and generality were seen by many mathematicians as the guiding principles. There were cracks in this intellectual foundation, and the next twenty years were to see many of these prejudices disappear.
In my own field of analysis there had been overblown expectations in the 1950s and 1960s that abstract methods could be developed to solve a large range of very concrete problems. The correct axioms and clever theorems for abstract Banach spaces or algebras would conquer the day. By the late 1960s, groups in France and Sweden, along with the Chicago school in the U.S., had developed entirely new methods of a very concrete nature to solve old conjectures and open new frontiers. The hope of abstract salvation, at least in its most extreme forms, was revealed as naive. Especially for problems of a statistical nature, hard tools needed to be developed. (One should note that in other areas of mathematics, abstract methods have had spectacular success in solving even very concrete problems. What this means for the future of those fields is now a topic of broad speculation.)
How fascinating it is to look back on this period and observe Benoit Mandelbrot. He was looking at pictures, drawing conclusions in many fields, and being largely ignored by all. He was outside every orthodoxy imaginable.
To understand Mandelbrot's contributions to science, one must first give up the tendency to find a disciplinary pigeonhole for every scientist. What should one call someone who works simultaneously in mathematics, physics, economics, hydrology, geology, linguistics... ? And what should one think of someone whose method of entry into a field was often to find puzzling patterns, pictures, and statistics. The former could not be a scientist, and the latter could not be science! But Benoit Mandelbrot was really doing something very simple, at least at the entry point to a problem: He was looking at the pictures and letting them tell their own story.
In the mid 1500s, Galileo peered through telescopes to find astonishing celestial features imperceptible to the human eye. In very much the same spirit, Mandelbrot used the most modern computers available to investigate phenomena not well studied by closed formulas, and out popped strange and unexpected pictures. Furthermore, he worked with the idea that a feature observed in a mathematics problem might be related to "outliers" in financial data or the observed physics of some system. Perhaps these rare events or outliers were not actually so rare at all; perhaps they were even the main feature of the system!
After getting his foot in the mathematical door, Mandelbrot would start the next phase of research, erecting a mathematical framework and doing the hard estimates. Try today to explain to the scientifically literate highschool student that the beautiful fractal pictures on a computer screen are not interesting, at least not to be trusted, and try asserting that the fractals arising in wholly different problems are similar due just to chance.
While the aversion to looking at pictures has faded, there is still confusion as to why Mandelbrot's early works on fractals, e.g., his book The Fractal Geometry of Nature, generated such wild popularity in the general scientific community. One does not see on every page the "theoremproof" methodology of a mathematics textbook. Furthermore, though one can easily find theorems and rigorous proofs in the book, the phenomena and pictures discussed may seem to a mathematician to be unrelated, because there is not necessarily an exact theorem to link any two of them.
What a poor world we would live in if this were the only permitted method to study the universe! Consider the plight facing a working biologist, where all data sets are dirty and causality difficult to determine. Should one demand a theorem in this situation? Should a geologist looking at rock strata search first for a theorem, when the formalism of multifractal measures might be more important? An old tradition in science is to seek first a description of the system at hand; this apparently simpler problem is usually much more difficult than is generally believed. Few doubt that Kepler's laws would have been formulated without his first seeking patterns by poring over reams of data.
Perhaps, however, the pictures studied by Mandelbrot arose randomly, and any connection to interesting science is just a coincidence. The Mandelbrot set M offers an instructive example. Despite twenty years of intensive research by the world's best analysts, we still do not know whether M is locally connected (the MLC conjecture), and progress on this problem has rather ground to a halt. This is now seen as one of the most central problems of complex dynamics, and the solution would have many deep consequences. The geometry of M is known to be devilishly complicated; M. Shishikura proved that the boundary has dimension equal to two.
We know today that the "Sullivan dictionary" provides many analogues between iteration of rational functions and the theory of Kleinian groups, but there is very much that remains open. For example, we do not know whether it is possible for either a Julia set or a limit set (of a Kleinian group) to have positive area unless it is thefull sphere. If all Julia sets from quadratic polynomials have zero area, then the Fatou conjecture on density of hyperbolic systems would be proven for quadratics. It is also known that MLC implies both the Fatou conjecture for quadratics and the nonexistence of certain (but not all) Julia sets of positive area.
Another example is furnished by the Brownian boundary that is the subject of Plate 243 of The Fractal Geometry of Nature. Arguing by analogy and examination of simulations, Mandelbrot proposed that the Brownian boundary has dimension 4/3 and serves as a model for (continuous) selfavoiding random walks (SARW). The 4/3 conjecture was only recently solved by the spectacular work of G. Lawler, O. Schramm, and W. Werner. Their proof relied heavily on the New processes called SLE that Schramm invented. We now know that SLE (8/3) represents the Brownian boundary. This also proves another prediction of Mandelbrot that the two sides of the Brownian boundary are "statistically similar and independent." One of the major challenges in probability theory is to prove that SARW exists, and the new conjecture is that it can be identified with SLE (8/3).
The study of multifractals is another area where Mandelbrot played a leading role. Through multiplicative measures with singular support were known in certain areas of Fourier analysis and conformal mappings, their fine structure had not been examined, and they were virtually absent in discussions of physical problems until the work of Mandelbrot. He was also the first to write down f ( ) in the form of normalized logarithms of large deviation probabilities.
The status of these problems may be open, but the beautiful pictures, now easily reproduced by the aforementioned highschool student, continue to fascinate and amaze. What we see in this book is a glimpse of how Mandelbrot helped change our way of looking at the world. It is not just a book about a particular class of problems; it also contains a view on how to approach the mathematical and physical universe. This view is certain not to fade, but to be part of the working philosophy of the next mathematical revolution, wherever it may take us.
American Scientist %
May / June 2004 %
A nanoreview by BH
The term fractal was invented by Benoit Mandelbrot in 1975.... For those who wish to trace the later development of these ideas, Mandelbrot has been republishing much of his own work in a series of volumes he calls "selecta." The latest of these, Fractals and Chaos: The Mandelbrot Set and Beyond ..., brings together 25 papers from the past 25 years. Many of them are related in one way or another to the famous inkblot figure to which Mandelbrot's name is now firmly affixed. Of historical interest are some early images of this fractal object, produced with a crude dotmatrix printer. A few items in the collection have not been previously published, and all are accompanied by feisty commentary.
BioMedical Engineering OnLine %
03 May 2005 % Alberto Diaspro (U. Genoa IT)
Benoit Mandelbrot has produced a comprehensive, wellpresented review of essential topics related to Mandelbrot set theory and applications. The last part of the title "The Mandelbrot set and beyond" fully describes its potential allowing the reader to navigate through pictures, hardtofind early papers and important and effective chapters on the historical background. All chapters are assembled in a way that the overall mix becomes a very well integrated source of knowhow and knowledge bringing the readers into the Mandelbrot set world. The spirit of the book is well summarized in a sentence on page 34: "When seeking new insights, I look, look, look, and play with many pictures. (One picture is never enough)." It is certainly true that in the last twenty years, mathematics has changed so deeply that to younger persons some chapter's stories might be simply incredible (p.36), as well, one should admit that after Mandelbrot's sets, initially describing trees, coastlines' shapes or allowing measuring the length of the Britain coast, and after the seminal book on "The Fractal Geometry of Nature" our way of looking at the world changed. Mandelbrot wrote: "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". I think our vision of the world, from the atom to the higher length scales, is still changing using those concepts clearly illustrated in the current Mandelbrot's book. Selected notes and papers make this book unique within the several books published on this topic. It is clear the touch of the author under all aspects: a touch of pure genius.
There are five main topics dominating the book, namely: Quadratic iteration and its Mandelbrot set Quadratic Julia and Mandelbrot sets; Nonquadratic iterations Nonquadratic rational dynamics; Kleinian groups' limit set Iterated nonlinear function systems and the fractal limit sets of Kleinian groups; Multifractal invariant measures Exponentially vanishing multifractal measures; Background and History. Cumulative bibliography is impressive and well done. It is clearly pointed out, following the pathway through the book, how fractal geometry played an important role in offering a quantitative tool in several areas. Circumstances and facts are put together also to bring important lessons for young scientists. The author made a serious and effective effort to realize a book that contains more than history, more than mathematics... it is a sort of ideal book for stimulating new ideas, new concepts, and new discoveries. So far, it is an excellent book also for supporting courses at University, PhD and Post doc level. Moreover, it is indispensable for scientists not only as a lesson of a pathway in science but also as an important source for science of tomorrow. This is a valuable reference source to researchers from these and related areas including bioengineering, biophysics, nanobiosciences and, of course, applied mathematics.
Estreams Electronic Revies of Science & Technology References % Oct 2004 pNA. % 7, (9) Index , % James A. Buczynski (Seneca at York Learning Commons)
Benoit Mandelbrot, an award winning, academic nomad in economics, physiology, physics and mathematics, is credited with changing the way scientists in many fields, look at the world. Crude computer generated pictures were used to bridge the philosophical chasm between the accepted methodologies of scientific inquiry in the 1970s and available computing technology. Although not his intent, he solved an ingenuity gap in adopting computing as a tool of inquiry. The tool was available, but there was an absence of creativity in using it. Visualizations of fractals captured the imaginations of scholars and challenged them to approach real world problems in new ways. Fractals: Form, Chance & Dimension ... and ...Fractal Geometry of Nature influenced many (sold over 200 000 copies) in spite of its deviation from the theoremproof methodology expected of mathematics books. Researchers today do not think twice about using a computer simulation to begin investigating a problem. Mandelbrot founded the influential fields of fractal and multifractal geometry.
Fractals are rough or broken geometric shapes that are indefinitely complex and are selfsimilar at different levels. In essence, fractals are nested shapes. The term "fractal," coined by Mandelbrot, is derived from the Latin word "fractus" meaning broken or irregular. Examples include: water channels in watersheds, and blood vessels and nerves in an organism. Although the theory of fractals existed long before computers, it could not be tested because the calculations were too arduous. Mandelbrot plotted the formula by computer and found to his amazement that fractal math was good at generating images of natural systems such as shore lines, capillary beds and erratic chance phenomena like the dynamics of turbulence. Nature, in essence, explained by mathematical formulas.
Fractals and Chaos: The Mandelbrot Set and Beyond is a mixture of previously published articles, commentary on those works and new material. The book is not specifically focused on the interconnections between fractals and dynamical systems but a look back at his contributions to the FatouJulia iterations of the quadratic map: z2 + C; FatouJulia iterations of other rational maps; Poincares Kleinian limit sets and related singular measures. Although the books content is marketed as widely accessible to nonmathematicians, its topical coverage clearly demonstrates it is geared towards those with at the very least, a strong background in undergraduate mathematics. Accessible content is widely scattered throughout the book, especially in chapter 17 and 23. Mandelbrots commentary on his work taken together with his nonmathematician explanations and digressions seem to be targeted at his critics who have often written about a lack of context and clarity in his writings. Taken alone, they do not offer the lay reader much.
The book is well populated with black and white graphical computer outputs of fractals and conceptual illustrations. Never before published illustrations from 1977 to 1979, concerning his nonquadratic map fumblings are included. The cumulative bibliography spans seventeen pages and includes both referenced sources ("many of which are ancient," as Mandelbrot states) and copyright credits. Recommended for all university science libraries. For all those disappointed readers out there, Mandelbrot (together with Richard Hudson) recently released a new finance focused work, The Misbehavior of Markets (Basic Books, 2004) that is specifically aimed at lay readers and has already got people talking. His celebrity continues.
Mathematical Association of AmericaMAA Online Book Reviews
% 3/9/2005 Mihaela Poplicher
This book contains early papers by Benoit Mandelbrot, as well as additional chapters describing the historical background and context. The material is grouped under five topics:
" I Quadratic Julia and Mandelbrot Sets
" II Nonquadratic Rational Dynamics
III Iterated Nonlinear Function Systems and the Fractal Limit Sets of Kleinian Groups
IV Multifractal Invariant Measures
V Background and History
Most of the papers included have been published before, beginning with the early 1980s until 2003, but there a few new ones. The work included in this book, "Selecta Volume C" was done by Mandelbrot while he was working at the IBM T. J. Watson Research Center and at Yale University. The book is dedicated to the memory of the author's uncle, Szolem Mandelbrojt, himself a mathematician who greatly influenced his nephew Benoit. The book also includes many illustrations, some of them very easily recognizable.
In his Foreword, Professor Peter W. Jones of Yale University notes: "It is only twentythree years since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot set. The graphics available at that time seem primitive today, and Mandelbrot's working drafts were even harder to interpret. But how that picture has changed our views of the mathematical and physical universe!" And later: "What we see in this book is a glimpse of how Mandelbrot helped change our way of looking at the world. It is not just a book about a particular class of problems; it also contains a view on how to approach the mathematical and physical universe."
In his Preface to the book, Mandelbrot emphasizes the fact that, although the book's main goal is to show the interconnections between fractals and chaotic dynamical systems, "this is neither a monograph on those interconnections, nor a textbook."
Of course the mathematical papers are extremely interesting, and a collection of all of them put together by their author is really a treat, but what I have found even more fascinating (and more entertaining to read, even for nonspecialists) are the papers dealing with background, historical notes, biographical notes, commentaries, etc. Most of these have not been published before, so there is no hope finding them in another place. I will mention just a few examples, leaving the readers to discover the others for themselves.
Chapter C1, "Introduction to papers on quadratic dynamics: a progression from seeing to discovering," has tantalizing sections such as "Computing at Harvard in 1980" and "The culture of mathematics during the 1960s and 1970s".
Chapter C2, "Acknowledgments related to quadratic dynamics", contains wonderful references and tributes to big names in mathematics: Nicolas Bourbaki, Andre Weil, Jean Dieudonn, Laurent Schwartz, Marshall Stone, Gaston Julia, and others (including Szolem Mandelbrojt, the uncle who was Mandelbrot's "earliest and Foremost mentor").
Chapter C15, "Introduction to papers on Kleinian groups, their fractal limit sets, and IFS: history, recollections, and acknowledgments", contains sections such as "The early history of Poincar's great innovation, one he chose to call 'Kleinian" groups'" and "Was the progress from pictures for their own sake, to open new mathematical vistas preordained?" and "The notion of IFS (iterated function system or schemes) or decomposable dynamical systems".
The last part of the book, "Part V: Background and History", is my favorite. Here is how the author introduces it: "Some chapters in this part are introductions whose aim is to assist even the nonexpert in gaining something from this book". And this is exactly what C23, "The inexhaustible function z squared plus c", C24 , "The Fatou and Julia stories", and C25, "Mathematical analysis while in the wilderness", are doing.
In summary, this is a wonderful book for a large group of readers: nonexperts interested in some introduction to Mandelbrot's work and biography, with historical notes and commentaries; as well as for specialists learning and researching in quadratic and nonquadratic dynamics, Julia and Mandelbrot sets, Kleinian limit sets, Minkowski measure. Reading this book was a pleasure.
The Mathematical Gazette %
March 2005
Ren L. Schilling (U. Marburg, DE)
Fractals, fractal geometry or chaos theory have been a hot topic in scientific research. It may come as a surprise that much of the theory as we know it was initiated during the last 30 years and by the vision of one man: Benoit Mandelbrot. The story starts in 1975 with Mandelbrot's small booklet Les objets fractals. Forme, hasard et dimension (published by Flammarion, Paris). Already in 1977 it was translated and expanded into Fractals: Form, Chance and Dimension (W.H. Freeman, San Francisco), but the breakthrough came in 1980 with the first picture of the Mandelbrot set in `Fractal aspects of the iteration of z ( z(1  z) for complex and z', Ann. New York Acad. Sci. and reprinted on pp. 3751 in the present volume. It was, however, Mandelbrot's 1982 masterpiece The Fractal Geometry of Nature (W.H. Freeman, New York) that popularized the subject. Mandelbrot's book is a scientific, philosophic and pictorial treatise at the same time and it is one of the rare specimen of serious mathematics books that can be read and reread at many different levels.
The volume under review is `Selecta C' of Mandelbrot's oeuvre. It is a selection of papers which appeared between 1980 and 2003, dealing with (non)quadratic rational dynamics, iterated (nonlinear) function systems and multifractal measures. Alongside some important and very technical original papers, there is a highly readable (also for the nonspecialist) introduction and surveytype original contributions, extracts from his 1982 monograph as well as unpublished material. The last chapter is devoted to a brief historical account of the subject's early heroes: Pierre Fatou and Gaston Julia. Rather than being a juxtaposition of papers, Mandelbrot succeeded in creating a readable selection of material which contains new original contributions. The papers featured in the book are sometimes corrected and annotated; that in this process the original pagination was lost is somewhat unfortunate. The style is what one could call `truly Mandelbrotian', a mixture of hard science, often with a personal touch, some (sometimes quite lopsided) personal notes and recollections and always the urge to convey a message.
A brief word on the numbering. Selecta C is a companion volume of [books] which have been appearing with Springer since 1997... Whether these are to be seen as (early) volumes of Mandelbrot's `Selecta', I don't know, but it would make perfect sense.
Mandelbrot has done it again: here is a book that will be as important for the scientific community, many of Mandelbrot's early papers appeared in hardtoget journal and proceedings volumes, as it will be appealing to a general informed audience.
Mathematical Reviews %Martina Zhle (U. Iena, DE)
This is not only the fourth volume of the SelectaSelected papers of B. B. Mandelbrot. It is also a philosophically colored report on the most important stations of his scientific life. Going back to the roots, enlightening the fruitful 1980s and concluding with some future challenges the author gives a deep insight into his complex work determined by "seeing and discovering".
In the last decades Benoit Mandelbrot has been a great stimulator and multiplier of mathematical ideas in sciences, medicine, economy and linguistics. Reviving and successfully using experimental mathematics he realized his old dream "of helping unscramble the messiness of nature". The present volume emphasizes the role that more and more powerful computer generations have played in working out the interconnections between fractals and chaotic dynamical systems. Without a doubt B. Mandelbrot is recognized as a founder of modern fractal geometry, and even now dynamical systems are one of the main sources for generating concrete models of fractal sets. Since his experimental discovery of the most exciting example now called the Mandelbrot set in 1979/1980, an increasing interaction between such computer aided results and deep mathematical theories has stimulated new and challenging developments.
The book starts with a Foreword by P. W. Jones, an essential promotor of Mandelbrot's early work at Yale University, and a recognized professor of mathematical analysis. It is historically interesting, and points out Mandelbrot's influence on some highlights in theoretical results of fractal geometry, dynamics and stochastics.
In the Preface the author sketches the four main topics of the book, his motivations and tools as well as relationships to the former three volumes of the Selecta.
The main parts are the following: Part I. Quadratic Julia and Mandelbrot sets. Here some basic articles from 19821985 are reprinted with new forewords and additional comments. At the beginning an "Introduction to papers on quadratic dynamics" and "Acknowledgements related to quadratic dynamics" are inserted. In particular, the early influence of Poincar\'{e}, Hadamard, Julia and Fatou on Mandelbrot's work is emphasized and the important related theoretical results of Douady and Hubbard are mentioned. (A comprehensible presentation of the last developments for a general audience may be found in [H.O. Peitgen, H. Jrgens and D. Saupe, Fractals for the classroom. Part 2, Springer, New York, 1992; MR1181421 (93k:00016)].)
Part II. Nonquadratic rational dynamics. The Julia sets and the Mandelbrot sets associated with some rational mappings are discussed. In particular, a computer assisted homage to the wave paintings of the Japanese K. Hukosai (17601949) is reprinted. Part III. Iterated nonlinear function systems and the fractal limit sets of Kleinian groups. Limit sets of Kleinian groups provide another large source for fractals with many analogues to the Julia sets of conformal dynamics. Mandelbrot's experimental discoveries in this field are illustrated by reprints of three articles introduced with some historical background. (Here we refer to the new book [D. Mumford, C. Series and D. J. Wright, Indra's pearls, Cambridge Univ. Press, New York, 2002; MR1913879 (2003f:00005)].) Part IV. Multifractal invariant measures. The study of multifractals is another area where Mandelbrot played a pioneering role. Though this was the topic of a former volume of the Selecta, Mandelbrot reprints here one of his later articles and two joint papers with Gutzwiller and Evertsz. The purpose is proclaimed in the Introduction: these papers related to dynamics deal with examples where the local Hlder exponents in the multifractal spectrum of the measures are unbounded. For the case of the physically relevant diffusion limited aggregation this is only a longstanding conjecture (cf. the paper with Evertsz). A strong mathematical model for DLA confirming the above statement has not yet been developed. In this context the end of the book may be understood as a challenge to researchers in related fields. Part V. Background and history. This appendix contains, in particular, some historical information about Julia and Fatou as well as on Mandelbrot's relationship to the Bourbaki group. Since his famous book "The fractal geometry of nature" in 1982 the ideas of fractals have penetrated into many fields of mathematics. Nowadays one speaks not only of fractal geometry, but also of fractal analysis or fractal stochastics. Nevertheless, the available theories are only cornerstones at the beginning of a presumably long scientific development. The Selecta of Mandelbrot should be considered as a basic contribution to force the interaction between pure and applied mathematics in this beautiful field with natural dissonances and harmony.
Nature % July 1, 2004, 430
% Kenneth Falconer (Pure mathematics,U. of St. Andrews, St. Andrews, UK)
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gdlH'%&dPgdrdinary complexity and beauty appeared in scientific and glossy magazines, on the walls of art galleries and classrooms, on posters and even on tablemats. With the increasing availability of personal computers, drawing the Mandelbrot set became a standard exercise for those learning programming, and it was frequently an addiction for computer buffs, who were able to explore its intricacy by forever homing in on parts of the structure. Perhaps it is not surprising that such a simple procedure enabling almost anyone to produce an object of immense sophistication and attractiveness caught the public imagination.
The definition of the Mandelbrot set, denoted by M, is indeed extremely simple. Given a complex number c, start at the origin 0 and follow the trail of points obtained by repeatedly applying the transformation f(z) = z2 + c, that is, the sequence 0, c, c2 + c, (c2 + c)2 + c... If these points never go far away from the origin then c is in M, but if they wander off to infinity, c is not in M. This straightforward check allows one to scan across a region of the complex plane to determine the extent of M.
Crude pictures of M show a main cardioid surrounded by circular 'buds' of decreasing size. But more detailed investigation, pioneered by Benoit Mandelbrot in 1980, reveals much, much more: the buds are all surrounded by smaller buds, which in turn support even smaller ones, and so on. Homing in on the boundary of M reveals a menagerie of multibranched spirals, dragons and seahorses. Hairs of imperceptible fineness extend from the buds, holding along their lengths minute replicas of the entire Mandelbrot set.
Is the Mandelbrot set just a pretty curiosity? Far from it. It is a fundamental parameter set that encodes an enormous amount of information about nonlinear processes. First, the position of a complex number c relative to M tells us a great deal about the iteration of the quadratic mapping f(z) = z2 + c. The (filledin) 'Julia set' at c consists of those complex numbers z whose iterates under repeated application of f(z) = z2 + c never wander far from the origin. This Julia set comprises a single piece precisely when c lies in the Mandelbrot set. (Interestingly, this topological dichotomy was noted by Pierre Fatou and Gaston Julia in 191819, but it was many years before its real significance and delicacy was appreciated.)
Much more than this, the exact position of c in M, such as the bud in which it lies, gives a very full description of how the iterates of f(z) behave: for example, whether there are periodic cycles. Even more surprising is that, although defined in terms of the simplest of nonlinear maps, f(z) = z2 + c, the Mandelbrot set is 'universal' in that it underlies the behaviour of very large classes of more complicated nonlinear mappings, the likes of which crop up throughout modern mathematics and its applications.
The emergence of the Mandelbrot set in 1980 led to a flurry of activity among mathematicians trying to understand its structure and significance, resulting in some of the most impressive advances in pure mathematics in recent years. In 1982, Adrien Douady and John Hubbard proved the (far from trivial) fact that M is connected, though it is still unknown whether M is locally connected can you travel between nearby points of M staying inside M without making too long a detour? In 1998, Mitsuhiro Shishikura showed that the boundary of M has fractal dimension 2, which means that it is just about as complicated as can be, though it is still not known whether this boundary has positive area.
This is the fourth volume of Mandelbrot's Selecta, comprising edited reprints of the author's papers. Largely from the 1980s, these include the series of seminal papers that revealed the magnificence and omnipresence of the Mandelbrot set, together with other papers related to the iteration of functions. Several sections provide an overview of the work along with its scientific and historical background. One chapter has been written specifically to help the nonexpert appreciate the rest of the book.
Much of the material does not require particularly technical knowledge, so the book should be accessible to a wide readership. It provides a fascinating insight into the author's journey of seeing and discovering as the early pictures of the Mandelbrot set started to reveal a whole new world. It gives a feeling for his philosophy and approach of experimental mathematics an approach that has changed the way we think about mathematics and science.
Pure and Applied Geophysics (PAGeoph) % 162, 2005 % Helmut Kirchner (U. ParisSud XI, FR)
Imagine you meet a great scientist in person, maybe over dinner at his house. He starts to remember how his discoveries came about, what little detail in his education brought him on the right track, what coincidence made him study this or that. There might be details nobody else will ever know; they might be too private or inconsequential, but surely the listener will remember them. After all, why somebody succeeds, why somebody does great work is difficult to grasp. Some people meet a quick, possibly undeserved success, others were, as one says, ahead of their time and went unrecognized.
On the other hand, in a more formal setting, historians of science write about influences, traditions and changes of paradigms. There are even Departments of History of Science busying themselves with similar approaches. Estates of scientists are bought by universities, papers get deciphered. What at the time was a polite letter might assume, in hindsight, ponderous and wonderous significance if found in an archive.
In Mandelbrots SELECTA volume C the reader gets both. A scientist becomes rarely the historian of his own work, however here it is the case. After three previous 1020 Book reviews Pure appl. geophys., selecta in the same style, the selecta E on finance, and the selecta N and H on fractals, this is the fourth and last volume of Mandelbrots chosen pieces. Hereafter further selecta will appear on the internet.
Mandelbrot comments upon each of the selected articles: how it came about, what it means, and what it really means. He explains why he did this and not that, and why he did not do something else. Essentially he claims that he was the first, since the ancient Greek geometers, to have introduced pictures into mathematics. A great achievement, a great first. After all, he comes from the French tradition of mathematical iconoclasts. In 1788, in the preface to his Mecanique analytique, the great Lagrange wrote On ne trouvera point de figures dans cet ouvrage, seulement des operations algebriques. The word algebraic instead of analytic is characteristic. In the twentieth century, due to the formalistic Bourbaki movement, some of French mathematics had turned somewhat stale, dry and infertile. Against this iconoclastic background, Mandelbrots iconic approach looks very bold and daring. (Once a Japanese friend of mine explained to me how he taught film theory at the University of Tokyo: Just look and think, that is what I tell my students)
Mandelbrot did not call his set the Mandelbrot set, that name was coined by Doujady. Consider the iterated mapping z z2+c, where both c and z are complex. If c is kept constant and z is allowed to vary in the field of complex numbers, the iteration traces a Julia set. If z is kept constant and c varies, the iteration traces a Mandelbrot set. Playing with his old fashioned IBM computer in 1979 or 1980, Mandelbrot looked carefully at the crude images obtained, and he saw wonderful things.
Chaotic and fractal work has become popular among the mathematicians. The first images gave rise to further and more serious mathematical work. In his book Mandelbrot generously shares his fame with his predecessors Julia, Fatou and Poincar. The introductions to reprinted classical texts, written by him specially for this volume, are more than anecdotes. They illustrate how a theory emerges, how answers create new questions, and how progress is made. The reader will be grateful not only for these historical explanatory remarks, but also for pieces adapted or written by Mandelbrot especially for this book. Collectively they give a compelling account of how a new branch of mathematics was created by the author. This delighful book makes good reading.
Times Higher Education Supplement %
Shaun Bullett (Queen Mary C., U London, UK)
Do mathematicians discover mathematics or do they create it? One has only to look at the intricate beauty of the Mandelbrot set to marvel at what was waiting to be found. But to explain its structure is an unfinished task that has taken some of the most creative mathematics of our time.
Benoit Mandelbrot was 80 last year, and it is 25 years since he first saw the ubiquitous set that bears his name. This is the collection of all complex numbers, "c", such that the repeated operation of squaring followed by adding c, beginning with zero, gives a sequence that is bounded. In 197980, Mandelbrot started investigating the values of c for which this sequence tends to a periodic cycle, and was astonished at the extraordinary structure that emerged in his computer plots. His pictures revealed a central "mainland" surrounded by a network of "islands", each a scaleddown copy of the mainland (these islands were removed from the first published pictures by a zealous printer who believed them to be "dust"!).
Adrien Douady named this structure the "Mandelbrot set". In 1982, Douady and John H. Hubbard proved that it is connected and formulated the conjecture ("MLC") that it is locally connected a conjecture that if ever proved would finally resolve its remaining mysteries.
This book is a selection of articles from the 1980s and early 1990s, together with previously unpublished material from the same period, linked by additional chapters containing commentaries and reminiscences. The papers are grouped under five main headings, from quadratic Julia and Mandelbrot sets, to the limit sets of Kleinian groups and invariant multifractal measures. They present Mandelbrot's personal account of his discoveries and his mathematical philosophy rather than a comprehensive guide to the subject.
Mandelbrot's work is driven by an insatiable curiosity in what he sees, guided by a mathematical eye strongly influenced by the heritage of Henri Poincar, who initiated the study of Kleinian groups and whose ideas about dynamics inspired the theory of iterated complex maps developed by Pierre Fatou and Gaston Julia. FatouJulia theory, a triumph of complex analysis, was neglected in the mid20th century  even by Mandelbrot as a student, despite his having Julia as a teacher at the Ecole Polytechnique. The arrival of the microcomputer brought a huge new flowering of theory, as well as experiments, in the 1980s, interacting with and complementing remarkable advances in nonlinear dynamics and hyperbolic geometry.
One does not need to be a professional mathematician to read these papers, though it helps to have a scientific background. They include very readable historical and polemical commentaries Mandelbrot certainly lets you know his views on the 20thcentury trend towards general theory and abstraction.
The numerous computer plots are pleasing to the eye, and the primitive early ones are of particular interest. Fractals were brought to a wider public by the superb colour graphics of HeinzOtto Peitgen and Peter Richter (The Beauty of Fractals, 1986). The recent book by David Mumford, Caroline Series and David Wright (Indra's Pearls: The Vision of Felix Klein), containing stunning pictures of limit sets of Kleinian groups, is also highly recommended by Mandelbrot.
The informal mix of mathematics and commentary in Mandelbrot's book provides a fascinating insight into his motivation and method. Often at odds with the mathematical mainstream, he reminds us of the value of images and of basing our mathematics on what we see. Mandelbrot looked at the world and saw fractals (a term he coined in 1975), helping to change how we view the mathematical and physical universe.
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B.B. MANDELBROT % SCRAPBOOK OF SELECTA BOOKS % September 29, 2006 % PAGE 2
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