1.1. Review by: Seymour Sherman.
Mathematical Reviews MR0057495 (15,238i).

The author undertakes to study problems of communication from the point of view of the mathematical theory of games of strategy involving three players: the sender, Nature, and the receiver, with various conditions among the players. He points out that the strategic framework can be used directly as in the work of Shannon (1948) on the technical problems of transmission given the statistical properties of messages and the physical properties of Nature. He asserts that, equally, the inverse approach can be used: one can construct perfect plays and show that the results are verified by physical phenomena. Two examples are considered in detail; thermodynamics as an optimal minimax game between sender and Nature and the statistical structure of language in relation to a minimum coalition of sender and receiver.

2.1. Review by: Philip Morrison.
Scientific American 233 (5) (1975), 143-147.

The book and in some sense the ideas begin with Jean Perrin, whose study of quantitative Brownian motion visibly displayed the reality of the kinetic theory before World War I. Perrin reminds us that for mathematicians, with their nice analysis, "curves which have no tangent are the rule, and the well-regulated ones, such as the circle, are very interesting but highly special cases." These views seemed only an "intellectual exercise, without doubt ingenious, but ... sterile. ... Nature presents no such complications and does not even suggest the idea. But the contrary is true, and the logic of the mathematicians has kept them closer to reality than the practical representations of the physicists." Even the great analyst Charles Hermite once wrote to Thomas Jean Stieltjes of turning away "in fright and horror from the lamentable affliction of functions which have no derivative."

Most appreciators of mathematics have long understood this point on the molecular level. What Dr Mandelbrot has shown is a world of visible form that is governed by the properties of Brownian motion, suitably generalised by the work of many hands, including his own original studies.

Take the analyst's "snowflake" curve (named here after its discoverer, H von Koch), which is made in an elementary way from an equilateral triangle of unit side by replacing the middle third of each side with a "cape," itself the two jutting equal sides of a triangle a third as large as the original, and so on, repeating indefinitely. With each repetition the length increases by the factor $\large\frac{4}{3}\normalsize$. Such a continuous curve has in the limit no tangents, infinite length and zero area.

The mathematicians have presented a number of measures applicable to such a remarkable curve that can distinguish it overall from a circle, even though it too is merely a one-dimensional manifold. Consider two properties of the snowflake curve. It is self-similar: magnify any section and you see under the magnifier just the same form on a smaller scale. Step off its length with a pair of dividers; with dividers set at one unit distance the length you measure will be three units. Next try a one-third-unit divider setting. The length is now four units. The stepped-off length will increase continually as the divider setting grows smaller, following a simple power law, with the calculated exponent .26.

A straight line shares some properties with the snowflake. It too is self-similar, but its stepped-off length is the same for every divider opening. It has a dimension of unity, in our usual way of thinking. Here arises the book's title: it is now plausible to assign the snowflake curve a generalised dimension (applicable to self-similar curves) that is greater than that of its tame kindred, a straight line. The general formula sensibly assigns a generalised dimension 1 to the straight line and "dimension" $\large \frac {\log 4}{\log 3}\normalsize = 1.26...$ . to the grander snowflake. Hence the book's title. Dr Mandelbrot offers us a new word, fractal, suited to the tongue in both French and English, to convey this meaning of intermediate dimensionality. ("Fractional" itself would grate if it were applied to a number that is in general not rational. )

Where is the physics? Here lies the novelty and the bulk of the book. Of course we have no physical snowflake curves. Nature gives no infinities, not even within molecular collisions. There is a cutoff at the angstrom level. Still, surprises abound. The land-sea shore lines of the real world approximate the snowflake curve! Two modifications are needed. There is no such formal regularity in the coast of Brittany; it is self similar in the statistical sense only. A map at a 10-kilometre scale shows some complex bays and peninsulas; so does a map at a 0.1-kilometre scale. On the average the forms do not differ (apart, of course, from any particular fishing cove we know). Empirically the stepped-off length of a coastline on maps at varying scales obeys a power law like the snow flake curve's, from a scale of hundreds of kilometres down to one of perhaps metres, where geography stops and pebbles begin. Only the exponent differs.

3.1. Review by: Clive W Kilmister.
The Mathematical Gazette 62 (420) (1978), 130-132.

This review is directed towards two distinct readerships. The first consists of the smallish band of mathematicians who are on everyday terms with Hausdorff dimension and who frequently meet sets of (misleadingly called) fractional dimension. To them I commend this as a handsome coffee-table book and ask them if they had realised before that some of that stuff was actually of practical use. While pondering this they should then skip to the last paragraph.

But Mandelbrot's contention, which he demonstrates admirably, is that these matters are much too important to be left to the fractional dimensionists. Just as Hille saw semi-groups whenever he went out in the street, so Mandelbrot sees fractals (from, he tells us as the inventor, Latin fractus: irregular or fragmented). Geometry can describe man-made objects but is useless in considering almost all natural ones (coastlines, boundaries of clouds,...); so Euclid is replaced as hero by a celestial committee of Weierstrass, Cantor, Peano, Lebesgue, Hausdorff, Koch, Sierpinski and Besicovitch, whose ideas have condensed into fractals under Mandelbrot's supervision. And, he emphasises, it isn't that we are suddenly to go over to considering the pathological cases; but to realising that they weren't odd after all. As Perrin said, "Those who hear of curves without tangents, or of functions without derivatives, often think at first that Nature presents no such complications, nor even suggests them. The contrary, however, is true and the logic of the mathematicians has kept them nearer to reality than the practical representations employed by the physicists." Perrin had in mind the Brownian motion; the curves are the best known fractals, and are typical in involving chance, but unusual in having been well understood for as long as fifty years.

There are several ways in to the author's thought: I found the most useful was in terms of dimension. Everyone knows, at least vaguely, that the old idea of "number of parameters needed to specify" got into difficulties somewhere in the turbulent half-century of mathematics centred on 1900. Out of the difficulties came not one but two definitions of dimension for an arbitrary set of points. The topological dimension of Lebesgue and Brouwer (1911 and 1913) - commonly attributed to its rediscoverers Menger and Urysohn in 1922-starts from Lebesgue's noticing that a square can be covered by small 'bricks' in such a way that no point is contained in or on more than three bricks. Similarly a cube in euclidean $n$-space can be decomposed into bricks, no more than $(n + 1)$ of which meet; and this procedure gives a definition of an integer-valued variable $D$ for any set which can reasonably be called its dimension. ...
...
The best reason for buying this book is probably the pictures, though, which are good enough to recommend it over the range from the first group of review readers, who have just started to read again here, all the way to the school library which wants to provide stimulating background and is not afraid of putting an occasional error before the young. To illustrate a remark of Perrin's about the white flakes that arise by salting a soap solution, there is a "fractal flake and its shadow" - not a photograph of a real flake but a computer-generated shape which illustrates the geometric characteristics embodied in Perrin's description. (In fact the flake is the volume of space in which a fractional Brownian space-to-line function exceeds a certain threshold, with a fractal surface of dimension 2.25.) Because any useful illustration of fractals needs a large number of very accurately placed strokes on the page, hand drawing would have been prohibitive; but the author wanted to produce a picture-book, and he was able to do this by a computer graphics device that produces camera-ready copy. The result is to be highly commended; whether or not fractals are as universal as they seem here, they are certainly sets of widespread occurrence. The failure to realise this has obscured several branches of science already: but there is now no excuse.

3.2. Review by: Freeman Dyson.
Science, New Series 200 (4342) (1978), 677-678.

"Fractal" is a word invented by Mandelbrot to bring together under one heading a large class of objects that have certain structural features in common although they appear in diverse contexts in astronomy, geography, biology, fluid dynamics of Newton. Modern mathematics began with Cantor's set theory and Peano's space-filling curve. Historically, the revolution was forced by the discovery of mathematical structures that did not fit the patterns of Euclid and Newton. These new structures were regarded by contemporary mathematicians as "pathological." They were described as a "gallery of monsters," kin to the cubist painting and atonal music that were upsetting established standards of taste in the arts at about the same time. The mathematicians who created the monsters regarded them as important in showing that the world of pure mathematics contains a richness of possibilities going far beyond the simple structures that they saw in nature. Twentieth-century mathematics flowered in the belief that it had transcended completely the limitations imposed by its natural origins.

Now, as Mandelbrot points out with one example after another, we see that nature has played a joke on the mathematicians. The 19th-century mathematicians may have been lacking in imagination, but nature was not. The same pathological structures that the mathematicians invented to break loose from 19th-century naturalism turn out to be inherent in familiar objects all around us in nature. We do not have to look far to find them. Human tissue, as Mandelbrot notes, "is a bona fide fractal surface... Lebesgue-Osgood monsters are the very substance of our flesh!"

Unfortunately Mandelbrot's book is fractally written. The main theme is clear and important; some of the digressions are unimportant and unclear. There are many illustrations, all of them computer-generated mathematical structures rather than pictures of natural objects. Some of these computer print-outs are beautiful, some are illuminating, some are obscure and poorly explained. ...

The reviewer particularly enjoyed chapter 11, a dense cluster of historical anecdotes. Mandelbrot has an affinity for eccentric characters, and his historical scholarship is meticulously exact. One of his finest discoveries is a book called Two New Worlds, published in 1907 by Edmund Fournier D'Albe and containing the first description of a fractally clustered universe. "It is," Mandelbrot says, "the kind of work in which one is surprised to find anything sensible. One fears attracting attention to it, lest the more disputable bulk of the material be taken seriously." Mandelbrot the scientific maverick finds in Fournier D'Albe a kindred spirit, while Mandelbrot the historian, in a fine display of irony, describes D'Albe's book in words that could also be used to describe his own.

3.3. Review by: Leon Glass.
Mathematical Reviews MR0471493 (57 #11224).

The term "fractal" was coined by the author to designate "a mathematical set or concrete object whose form is extremely irregular and/or fragmented at all scales". This book reveals his fascination in rooting out fractal sets in nature and in finding mathematics appropriate to study them.

A familiar example of a fractal set taken from physics is the trajectory of a particle undergoing "classical Brownian motion". Here, as J Perrin pointed out in the beginning of this century, the path of the particle is so irregular that the notion of a velocity (or a derivative) does not make sense. If the trajectory is examined increasingly closely its length increases without bound. Moreover, although the trajectory of the particle is one-dimensional in the topological sense, the trajectory will practically fill the plane so that in the alternative sense of Hausdorff and Besicovitch the dimension of the trajectory is 2. Formally, a fractal set is defined by the author as "a set for which one has Hausdorff-Besicovitch dimension > topological dimension". A rigorous mathematical study of Brownian motion has been undertaken in the works of Wiener and Lévy. Mandelbrot shows how generalisations and extensions of Brownian motion can be used in diverse fields, for example, to describe geomorphological processes such as geometry of coastlines. The statistical geometry of the coastline can be analysed in the context of fractals.

The author delights in proposing that fractal sets, which many have considered in the past as mathematical "monsters", can be used as models of physical systems. For example, a space filling Peano curve is introduced "if a river together with its tributaries is to drain an area thoroughly, it must, so far as other constraints allow, penetrate everywhere". Cantor sets enter naturally as mathematical models of noise characteristics of transmission lines. Other applications of fractal sets to problems in stellar distribution, vascular geometry, turbulence and polymer geometry are described. Throughout, there is a fresh and insightful use of fractal notions to study what at first thought seem to be recondite or unapproachable geometrical concepts.

The book is beautifully produced. There is a large number of computer-generated illustrations of fractal sets, which are both aesthetically pleasing and enormously useful in understanding the geometric properties of fractal sets. The writing is clear and escapes the bounds of scientific prose. The book is fun to read. Formal mathematics are relegated wherever possible to a single chapter at the end of the book. Nonspecialists in set and probability theory will be able to follow the main ideas of the argument.

In addition to discussing the use of fractal sets as mathematical models of physical processes, the author gives a good deal of historical and biographical information concerning development of fractal thought. The reviewer was struck by the way many of the ideas developed in the text have been incompletely studied by earlier workers, and have been largely ignored or forgotten since. But the coherent viewpoint that the author imposes on irregular geometries seems to lend a unifying concept to a wide range of difficult problems. The reviewer expects this book will receive widespread circulation among mathematicians and natural scientists, and will have a significant impact on the study of the geometry of nature.

3.4. Review by: I J Good.
Journal of the American Statistical Association 73 (362) (1978), 438.

A potboiler this is not. At a time when most books on statistics are written with a pot of glue and a pair of scissors, it is a relief to find one written with daring originality.
...
The illustrations, many of which were produced with the help of experimental IBM equipment, are excellent, especially the Sierpinski sponge which puts most non-mathematical abstract art totally in the shade.

One of the more convincing applications discussed by Mandelbrot is to the meaning of the length of a coastline (modelled by a random fractal set). The length depends very much on the size of the step taken to measure it. When the length is plotted against the step length on double logarithmic paper, an approximate straight line is obtained, as was observed by, Lewis Fry Richardson in 1961 (unpublished). Mandelbrot unearths this work and shows its relationship to fractal dimensions. The fractal dimension $D$ of a coastline is related to its roughness and to its fragmentation, so that $D$ might be a convenient parameter in geography and geology. It might turn out to be related to other geological features and, if so, it would have considerable scientific interest.

The book is based on a French 1975 edition and on numerous publications by the author. In fact, the bibliography of some 400 items lists 46 by Mandelbrot. I like people who write for glory and not just for money.

Mandelbrot adopts the strategy of starting with elementary methods and delaying even the definition of fractal dimension to the end. This policy of losing one's audience one at a time instead of all at the same time is one that should always be adopted, especially in colloquia. Another good device used by Mandelbrot is to number the diagrams by the pages where they appear.

3.5. Review by: Denis Bresson.
Leonardo 12 (3) (1979), 248-249.

A string, a sheet of paper, a ball are respectively one-dimensional, two-dimensional, three-dimensional. One speaks about the length of a string, the surface of a sheet of paper and the volume of a ball. However, how does one deal with the surface of a sponge and the length of a piece of coastline that is highly irregular? Is a string still one-dimensional when seen through a microscope? A new definition of dimension should take into account the character of the irregularities occurring in shapes, in textures, etc.

Clearly, the irregularities of a coastline on a map of small scale are smoothed out, but, as the scale is increased, more and more irregularities will be depicted and there will be some similarity between the irregularities, and, if each part of the coastline, statistically speaking, is similar to the whole, then the coastline is said to be self-similar. Furthermore, if the coastline has extremely fine irregularities, the line depicting it takes on nearly the appearance of a surface. The dimension of such a line can be defined as being more than 1 and less than 2, if the concept of a fractal dimension is introduced.

Mandelbrot points out that most of the shapes in the real world can be considered in terms of a fractal dimension: drainage basins of rivers, nets of blood veins, lung alveoles, stellar matter, noise records, turbulence of fluid motion, etc. By means of randomised computer programs, he generates shapes of given fractal dimensions such as Brownian 'island' landscapes and archipelagos. So the concept of a fractal dimension is useful to explain shapes occurring in the real world and for making visual and auditory artworks.

The book has been written so as to be comprehensible to non-mathematicians. Because of the wide range of application of the concept and of the numerous illustrations in the book, it should also be exciting to visual artists who enjoy viewing the world from a fresh perspective.

3.6. Review by: Mark Kac.
American Scientist 66 (2) (1978), 250.

This remarkable book is reminiscent in spirit and appearance of D'Arcy Thompson's On Growth and Form and Hugo Steinhaus's Mathematical Snapshots. The central theme is that geometrical objects whose fractional dimension exceeds their topological one (fractals) and which hitherto have been purely mathematical curios are uniquely suited for describing a staggeringly large number of spatial patterns presented to us by nature. These patterns include the coast of Britain, clusters of stellar matter, meteorites, moon craters, soap, and sponges, to mention but a few.

One of the author's aims is to show how using a few relatively simple theoretical hints (mainly based on the concept of self-similarity), one can generate (by a computer) patterns (deterministic and random) which are uncannily similar to those found in the real world. The suspicion that there is more to all this than just playing a game is supported in part by a number of interesting purely mathematical theorems that have been suggested and subsequently rigorously established.

But even if this were not so, the new and fascinating world of patterns presented here is all the justification this book needs. Read it, or better yet, look at the pictures. It is inconceivable that it would fail to give you some food for thought.

3.7. Review by: Marius Iosifescu.
International Statistical Review 47 (3) (1979), 299.

The topic of this unusual book - a much modified and augmented second version of Les objets fractals: forme, hasard et dimension (Flammarion, Paris & Montreal, 1975) - is a class of sets for which the author coined the term 'fractals'. (The Latin adjective fractus means 'irregular or fragmented'.) To be precise, a fractal is a subset of Euclidean space whose Hausdorff-Besicovitch dimension exceeds its topological dimension. Many of the pathological curves and other 'monsters' of mathematical analysis - Weierstrass functions, space-filling curves, snowflake curves, Cantor sets - are fractals. The same is true of the trajectories of ordinary Brownian motion. It is perhaps most surprising fractals prove to be of use as 'standard' models of the irregularity and fragmentation constantly observed in Nature. A few examples of natural fractals are seacoasts, river networks, pulmonary membranes, intermittency, turbulence, clouds, star clusters, and word frequency. As the author puts it 'the very same properties that cause Cantor discontinua to be viewed as pathological turn out to be indispensable in a realistic model of intermittency'.

The author's approach melds analysis, probability and topology. The book proceeds in the manner of a scientific Essay. The mathematical treatment is informal but backed by adequate references to rigorous proofs. The illustrations are not photographs or drawings but striking computer-simulated shapes predicted by the theory (some of them resemble photographs taken by the crew of Apollo XI). There are 12 chapters as follows: I. Introduction; II. How long is the coast of Britain?; III. Uses of non-constrained chance; IV. Fractal events and noises; V. Fractal clusters of stellar matter; VI. Turbulence, intermittency and curdling; VII. Meteorites, Moon craters and soap; VIII. Uses of self constrained chance; IX. Fractional Brownian facets of rivers, relief, and turbulence; X. Miscellany; XI. Biographical and historical sketches; XII. Mathematical lexicon & addenda. (This reviewer wants to draw the reader's attention to the notice on Louis Bachelier (1870-1946) in Chapter XI who unquestionably appears to be the first to describe in detail most of the results of the mathematical theory of Brownian motion.) Mandelbrot's work belongs to that extremely rare kind of book in which one can read about a new and important idea.

3.8. Review by: Howard C Howland.
The Quarterly Review of Biology 53 (2) (1978), 216.

It would be too much to ask that a treatise on crooked lines have a straightforward development, and Mandelbrot's essay is every bit as tortuous as the lines upon which he discourses. The book is not without charm, but occasionally one feels one is reading the first draft of a master's thesis with many revisions to go. The approach is definitely not a practical one and is geared primarily to appeal to the type of professional or amateur mathematician who enjoys mathematical recreations. However, the book should certainly be examined by those interested in mathematical aspects of biological structures, if only to convince themselves that this is not the path they wish to follow. Given the size and elegance of production, the price is very reasonable.

4.1. Review by: J W Cannon.
The American Mathematical Monthly 91 (9) (1984), 594-598.

Benoit B Mandelbrot, in his book entitled The Fractal Geometry of Nature, systematically sets about convincing the reader's eye and mind that many of Nature's apparent irregularities can be efficiently and beautifully modelled by mathematical objects, some deterministic and some with a random component, but all highly irregular or fragmented, which Mandelbrot chooses to call fractals.

Each newly created computer-generated illustration makes the argument more compelling. On the rear jacket of the book, a Brownian earth that never was rises over a moonscape that never was, both created by the computer artistry of one of Mandelbrot's co-workers at IBM, Richard F Voss. The programs are based on simple geometric invariance principles. More recent programs create mountain ranges, valleys, rivers, clouds, and mists, all visually compelling, totally fake, again based on a few simple invariance principles.

Mandelbrot suggests tentative first fractal models for coastlines, galaxy clusters, turbulence, island clusters, trees, the lungs, Brownian motion, drainage systems, mountain ranges, irregular textures, error bursts in data transmission, stock market variations, and many more. The book is a rich source of beautiful pictures, interesting new mathematical models, and imaginative new terminology.

Mandelbrot presents a wide range of experimental evidence by other scientists to support his views that standard models fall short of explaining the same phenomena with the same efficiency, clarity, and adherence to fact. Mandelbrot describes the book as a manifesto and a casebook. It is full of strongly held opinions and claims of priority, full of historical anecdotes, apt illustration, and the best in computer art.

4.2. Review by: Mark Nelkin.
Bulletin of the American Meteorological Society 65 (10) (1984), 1112.

This is a revised and updated version of the author's 1977 book, Fractals: Form, Chance and Dimension. Since that time, the subject has become immensely fashionable in science. For those who don't yet know what a "fractal" is, this book is an interesting and entertaining place to find out. Despite its length, the author correctly describes it as "a scientific Essay because it is written from a personal point of view and without attempting completeness." This intensely personal style is the strength of the book and of the author, but it can sometimes be frustrating to the reader. The author has made an important synthesis over the past two decades. He remains extremely conscious of his earlier difficulties in persuading the scientific community of the value of his point of view. If you are looking for a systematic introduction to the concepts of this fascinating subject, you will not find it here or anywhere else. So far, the subject belongs to Benoit Mandelbrot, and you will have to be introduced to it in his highly individual style. You will surely not be bored. At the very least, you will see the ideas illustrated with some excellent and often aesthetically satisfying computer graphics.

The simplest example of a fractal is the Cantor set, which was once thought of as a mathematical curiosity of little practical importance. Consider the interval from zero to one on the real line. Remove the middle one-third. Take the two remaining end intervals and remove the middle third of each of them. Continue this process ad infinitum. The remaining set of points has topological dimension zero, but is uncountably infinite in number. It is useful to associate with this set a similarity dimension $D = \large \frac {\log 2}{\log 3}$. The set has the important feature of having the same structure at every scale except for a factor defining the length scale at which observations are being made. As described by Mandelbrot, mathematicians of the early twentieth century created many such self-similar structures that can be conveniently described in terms of effective dimensions which are not integers. He illustrates many of these with elegant computer-generated illustrations.

The process which recursively generates the Cantor set can be thought of as a cascade in which the interesting features occur because of an infinite number of iterations of a simple basic step. As a model for the onset of turbulent flow, there has been considerable recent interest in taking this basic step as a map of the real line onto itself. In Mandelbrot's book, this is generalised to consider quadratic maps of the complex plane onto itself. Certain properties of large numbers of such iterations can generate amazing geometric patterns. One of these is illustrated on the jacket cover. We are reminded how little we understand about even the simplest nonlinear operations when iterated many times. We begin to see why large scale computation and modern computer graphics are becoming so important to basic mathematics.

Although deterministic structures of fractional dimension are of considerable mathematical interest, the essential element for the connection with natural science is the addition of randomness. This was Mandelbrot's basic contribution. He combined the probabilist's view of Brownian motion with the geometry generated by self-similar cascades. The mixture of randomness and geometry which results is able to describe a wealth of natural phenomena. An early example is the length of a coastline. This typically grows as the length of meter stick used to measure it shrinks. If measured with arbitrary precision, the length of a coastline is effectively infinite. Mandelbrot tells us how to associate this phenomenon with a dimension greater than one, and how to generate such structures by computer simulation. Except for a basic scale factor, these structures are again the same on all length scales, but now the similarity is statistical. Particular realisations can be very different. The book shows many "natural" phenomena generated by computer, including coastlines, river basins, mountain ranges, and galaxy clusters. The illustration on the back of the jacket is particularly striking.
...
Benoit Mandelbrot's great passions are geometry and history. His book is replete with fascinating historical episodes, and biographical sketches of famous and not-so-famous people who have made important contributions. His book is not easy to read from cover to cover, but it is an interesting and entertaining introduction to an important scientific field which he rightly claims to have largely developed himself. I would have preferred that he had spent less time re minding us of this, but I enjoyed the book very much and recommend it highly.

4.3. Review by: Don Chakerian.
The College Mathematics Journal 15 (2) (1984), 175-177.

In 1975, Benoit Mandelbrot coined the term "fractal" to describe sets whose Hausdorff-Besicovitch dimension is larger than their dimension as defined in topology (the topological dimension of the fractal in the preceding example turns out to be 1). In numerous articles and in previous editions of the book under review, Mandelbrot has initiated the systematic application of fractals to the description of natural phenomena. He takes pleasure in finding instances of fractal behaviour in nature and in pointing out how sets previously viewed by mathematicians as monstrous or pathological can serve as accurate scientific models in diverse fields including fluid mechanics, geomorphology, cosmology, economics and linguistics. Peano curves, for instance, can play a role in modelling watersheds, botanical trees, and the human vascular system. Variants of the fractal we described above, dubbed by Mandelbrot the "Sierpinski gasket" (Mandelbrot enjoys baptising his newly domesticated monsters with appropriate names), give insight into the turbulence of fluids.

The present volume is an updated and greatly expanded revision of the author's earlier Fractals: Form, Chance, and Dimension (W H Freeman and Company, 1977). Almost all the beautiful computer-generated figures of the previous edition appear here again, often in improved form, with numerous additional figures. The middle of the book contains a brief "book-within-the-book," with a sampling of fractal lore illustrated by a stunning sequence of extraordinary colour plates. Particularly striking are the multicoloured "self-squared fractal dragon" and what at first sight appears to be a colourful view of the earth rising over a lunar landscape but on closer examination turns out to be an alien creation of the computer's mind. In Mandelbrot's words "... the art can be enjoyed for itself."

The material has been reorganised for easier assimilation. The initial part of the book is devoted mainly to fractals of a regular nature having the intriguing property of "self-similarity." ...
...
As in the previous edition, one finds at the end of the book a collection of fascinating biographical and historical sketches related to the development of fractal geometry, addenda containing most of the mathematical technicalities required for a rigorous treatment of the concepts discussed in the main body of the text, and an extensive bibliography (which has now grown to over 600 references). The book has been splendidly produced, and the revision has brought with it only a few typographical errors, all of an insignificant character.

Mandelbrot's writing is lively, stimulating, and provocative. We are taken on a personal tour through a museum of science (or, as some mathematicians of yore complained, a gallery of monsters), with the guide pointing gleefully to this or that exhibit and recounting enthusiastically the labours expended in bringing it to the public eye. Indeed, the author shuns the muted and impersonal tone of traditional scientific discourse and without false modesty lays claim forcefully to those things he was the first to either discover or appraise as valuable. However, the prospective reader should be forewarned that a thorough understanding of this exposition requires hard work. While the material is presented in such a way as to be accessible to as wide a scientific audience as possible, one often finds clarity sacrificed for the sake of art, and the smooth and charming prose sometimes disguises an ambiguous formulation whose interpretation is not straightforward. ...

Any shortcomings this work may have are outweighed by the striking and thought-provoking computer art and the wealth of information the book conveys about geometry and the geometric aspects of various sciences. Mandelbrot has shown how some long-neglected but fascinating parts of geometry can help us look at nature with a new eye and perceive order in apparent disorder.

4.4. Review by: R P Burn.
The Mathematical Gazette 68 (443) (1984), 71-72.

This book is a revised edition of Fractals: form, chance and dimension which was reviewed by C W Kilmister in the June 1978 issue of the Mathematical Gazette 62 (420), 130-132. I warmly commend Kilmister's review, every sentence of which applies equally well to this edition. The material has been somewhat rearranged and the computer drawn illustrations have been substantially enhanced making this an even more fascinating book to glance through.

The most serious defect of the earlier edition unfortunately remains. The fundamental concept of the book, Hausdorff dimension, is introduced in too abstract a way for an average undergraduate to absorb at a first reading. It would have been possible to provide a concrete approach to this unfamiliar concept had the raw numerical data of Richardson's empirical studies of coastlines and boundaries been provided instead of a log log graph and an algebraic summary of his results.

4.5. Review by: Serge Dubuc.
Mathematical Reviews MR0665254 (84h:00021).

This book is a substantial revision of an earlier one [Fractals: form, chance and dimension, 1977], which itself was an expanded version of the French original. Many people, scientists and others, were very much excited by these works. Many patterns of Nature are irregular and fragmented. There is a need for a mathematical theory of such forms. The author responds to this challenge by using fractal sets, sets whose Hausdorff-Besicovitch dimension is bigger than their topological dimension. Fractional dimension, self-similarity or statistical self-similarity and power laws are the main mathematical tools of this book. Using these devices, many physical or even biological systems are studied.

In the first chapters, paradoxical curves (Peano and Koch curves) are used and generalised in order to describe situations like geographical coastlines, trees, shape and distribution of islands. Cantor sets are presented as models of noise. Some other fractal sets are defined in order to deal with situations like galaxy clusters, geometry of turbulence, percolation in lattices, ramification. The second half of the book introduces random fractals, mainly but not exclusively through the Brownian motion. These last models are then applied to river discharges, relief, texture, moon craters and so on. Coastlines and galaxy clusters are revisited. In all these cases, value of a dimension exponent is proposed and in some cases, an overall appearance is discussed. Very convincing computer-generated figures are companions to the text. Some power laws that are discussed come from Korcak, Pareto and Zipf.

Here are some main differences with the previous edition. The chapter on Koch curves is more elaborate. As an example, it is shown that one can fill the interior of a snowflake with another Koch-type curve. In other chapters, less known works of Fatou and Julia are recalled. Their very nice articles on iteration of functions have shown that many natural invariant sets under a nonlinear transformation have a very irregular boundary. This fact is important for the study of attractors for physical dynamical systems. In another direction, the author brings back a Poincaré chain of circles and constructs a new self-inverse fractal, inversions being with respect to circles. New chapters concern control of texture and more realistic models of galaxy clusters. Among 600 references at the end of the book, a hundred of them are publications since 1977. There is also a colour section of computer-generated fractals: eight new plates in colour highlight the book. Many people will enjoy this book.

4.6. Review by: Scott Ferson.
The Quarterly Review of Biology 58 (3) (1983), 412-413.

As exciting intellectually as it is beautifully visual, this work is a completely original synthesis of a modern geometry that escapes Euclid's notions and appears capable in large part of describing the highly irregular morphologies in nature. By way of definition, a fractal is a curve that although continuous is not smooth at any point. In the limit, such curves typically have infinite length, are nowhere differentiable and, although extraordinarily complex, display a certain self-similarity throughout all scales. Mandelbrot describes these objects and shows fairly convincingly that they are ubiquitous in nature and in our ideas about natural things. Examples include coastlines and stock-market price trajectories. Unrelated mechanistically, both examples exhibit pronounced - and characterising - irregularity. Along with curves, there are higher-dimensional fractals, including, according to Mandelbrot, mountainscapes and (botanical) trees, and lower-dimensional ones like hierarchically clustered point sets.

The prose is idiosyncratic but dependably interesting. The author's neologisms are numerous but etymologically sound and lexically prudent. Since it is to some extent a popularisation for the general scientist, there is only limited mathematics in the book. For pedagogical reasons, the models and mechanisms that generate fractals are neglected, and there is no discussion of occupational procedures. My only objection is that the book is perhaps too readable, that it concentrates inordinately on phenomenology. There are many references, however, and Mandelbrot heartily encourages further study.

There is not a lot of biology in this book, and this is a bit ironic since the archetypal fractal, Brownian motion, is named for a botanist. There are some biological applications, however, and it seems there is much room for progress in this direction. The complication arising from manifold iteration of even simple ontogenetic rules is frequently fractal in character. Mandelbrot suspects that the nature of the genetic code and its consequent pleitropy implies fractal complexity in biological form. Besides morphological fractals like tree branching and vascular systems, there appear to be other biologic structures and phenomena to which this geometry may apply. These include insect flight trajectories, phenetic clustering of evolving species and population size fluctuations. "Strange attractors" which cause dynamical chaos in deterministic models in ecology and physiology are fractal sets. The notion of a fractal is as primitive and profound as the Euclidean idea of the line and thus its utility ought to be about as wide-ranging.

4.7. Review by: Dane R Camp.
The Mathematics Teacher 89 (3) (1996), 256.

Benoit Mandlebrot's book The Fractal Geometry of Nature is a classic work that anyone with a serious interest in fractal geometry should have in her or his library. Originally published in 1977 and updated in 1983, it gives a unique historical insight into the growth of an idea and its diverse applications. However, it is not an easy read for students, who should be steered toward more popular treatments of the subject.

As the founder of fractal geometry, Mandlebrot admits that the book is meant to be a casebook as well as a manifesto and, there fore, "tends to digressions and interruptions." His casebook is organised into articles of varying length, difficulty, and depth. Each article can essentially stand alone, although some are almost conversational whereas others seem unnecessarily complex. Interestingly enough, across the book is a kind of repetition reminiscent of fractals themselves. Included in the book are topics familiar to "fractophiles," including the length of Britain's coast line, Koch curves, Cantor dust, and Sierpinski gaskets. But other items, such as the fractal dimension of the mammalian brain, pique the interest of readers who are used to the standard fare. Biographical and historical sketches are included, along with a wealth of black-and-white and colour diagrams.

Overall, I recommend this book for its importance in the history of fractal geometry and its value as a reference tool. But again, I caution that it is not light reading, as are more popular books, and is not designed to be a classroom textbook.

4.8. Review by: James N Boyd.
The Mathematics Teacher 76 (4) (1983), 288.

The poet tells us that Euclid gazed "on Beauty bare." Through the pages of this amazing book, the reader has set before him another sort of geometrical beauty - not bare or stark but clustered, convoluted, emerging from disorder and randomness.

A fractal in the plane is a curve so wildly abrupt in its changes in direction that the ordinary single dimension of length is insufficient as a measure of the curve. The curve takes up such a significant part of the plane that it must be assigned an effective dimension greater than one (its topological dimension) but less than two (the dimension of the plane). This nonintegral value of the effective dimension suggested the term fractal, which was first used by Mandelbrot.

One of the most striking features of a fractal curve is self-replication in all its parts down the scale from the gross toward the infinitesimal as smaller portions of the curve reproduce the shapes of larger portions. Such curves may suggest unusually beautiful patterns with the scaling property forging a recognisable symmetry. However, the most useful fractals have a random or statistical character at the roots of their structure. Applications have been found in the study of the flow of thermonuclear plasma, in the spectrum of the sun, in Brownian motion, in the modelling of coastlines, and even in economics and music.

The objects about us in the real world are not the objects of Euclid. Mandelbrot believes that the geometry of the world is fractal in nature with a beauty different from that of Euclid, but equally as satisfying.

4.9. Review by: I J Good
SIAM Review 26 (1) (1984), 131-132.

This book is a rewritten version of the author's Fractals: Form, Chance, and Dimension. The review of that book applies in large part to the present book which, however, is much larger, and contains many additional beautiful illustrations. For the benefit of those who find it inconvenient to consult the previous review, we repeat its first sentence, "A potboiler this is not." The author describes the book as "a casebook and a manifesto."

To put it roughly, a fractal is a configuration that looks much the same as its parts, magnified up, and sometimes having a hierarchical structure; for example, a cloud, a pattern on marble, a bubble-bath, Swiss cheese, a coastline, a turbulent mass of water, Weierstrass's nondifferentiable continuous function, and the set of all decimals on (0, 1) whose digits have the relative frequencies $p_{0}, p_{1}, ..., p_{q}. ...$

To judge by some of the illustrations the fractal number seems to give a good indication of the intuitive texture of a fractal geometrical configuration whereas the Lebesgue measure fails to do so because it is usually zero.

Most of the natural examples are terrestrial but a cosmological and hierarchical example is also given concerning our observable universe. On an even grander and more metaphysical scale perhaps the whole of nature is a hierarchical fractal of an infinite number of universes in which each one is a rotating black hole in another universe, as speculated in Physics Today (July, 1982, p. 15).

The title of the book emphasises that natural shapes often resemble fractals more than they resemble the figures of Euclidean geometry.

As the author mentions, a number of previous writers have recognised the interest of fractals, but Mandelbrot has devoted much of his life to them. He has produced a work of genuine originality.

4.10. Review by: Philip Morrison.
Scientific American 48 (3) (1983), 38-39.

In 1975 there appeared a small but dense and delightful volume in French around the neologism "fractal," the invention of this gifted, playful and tirelessly imaginative author. That work, "macedoine de livre," was reviewed in these columns with the enthusiasm it deserved. It was promptly enough served up in English, the key word still visible, the book fuller, much more visual, extended in scope and form. Here it is once again, the new edition straightforward of title, bearing a set of colour plates that exhibit the virtuosity of today's programmers, the mixture as fragrant and fresh as before.

The new form, perhaps increased a third in size, is ready for lucky readers who have not yet found their way to fractals, no less than for those who simply want more. Meanwhile the topic has become all but trendy. Strange attractors and renormalisation groups on a lattice, topics that blaze in the physics colloquiums these days, are here, full of evocative content albeit incomplete. The subject matter of the older editions is not neglected: the simplest idea behind the concept leads still to the understanding that the coastline of Maine has a length dependent on the inquirer. A ruler laid on the map from Portland to Calais gives a minimum answer; a line drawn through the indentations of bays and headlands on a highway map gives a longer one; the ruler put down centimetre by centimetre at real tidewater gives a much longer one, including every cobble and pebble.

The point is made quantitatively and is related to the notion of dimension. Such an exquisitely tortuous curve falls between the dimensionality of a line and that of a plane, reasonably enough. Hence the idea and the word "fractal." Of course, chance must enter, although rich deterministic examples are found in the dizzying idealised constructions of the long-known theory of real variables. Let the matter rest there; those who wish can find their way to the subject easily.

The book is no text, no set of definitions, no series of applications. It is a flowing source, as once the wonderful On Growth and Form by D'Arcy Wentworth Thompson came to us all. It too belongs on the small shelf of books that disclose the forms of nature. Thompson, classical in approach, centres on the simple forms hidden in the complex web of life. Mandelbrot, resting squarely on the mathematical moderns - Cantor, Koch, Hausdorff, Paul Levy - opens instead a path to the simplicity of plenitude. His heroic motifs are not spheres or spirals but intricate ideals of islands and reefs, stars and curdled milk, turbulent spray and pink noise, the abstract forms of nature's generative. profligacy. Naturally enough, the book itself is less lucid, but it is unifying, recursively fascinating, agreeably personal with out posturing and buoyant with scientific hope.

4.11. Review by: Colin Sparrow.
Journal of the Royal Statistical Society. Series A (General) 147 (4) (1984), 616-618.

Fractals, fractals, everywhere: A new mathematical and philosophical synthesis?

Mandelbrot's latest essay is intended, in the author's own words, as "a manifesto and a casebook". The thesis of the manifesto is summarised in the title; Mandelbrot insists, throughout the essay, that an understanding of fractal geometry is essential to the comprehension and description of various natural objects and processes. The various case studies, some of which will be mentioned later in this review, are taken from a wide range of different scientific disciplines and are all intended to illustrate this thesis. The essay is amply illustrated with computer generated fractal shapes, many of which bear an uncanny resemblance to well-known natural objects, and the reader cannot fail to be delighted by the colour plates of computer generated planetscapes; if seeing is believing then there is indeed a fractal face to the geometry of nature.

Mandelbrot's first case-study, and an ideal one with which to illustrate the idea of a natural fractal, is the question, "How long is the coast of Britain?" The answer to this question depends on the resolution of the measurement; the length as measured on a small scale map will be less than the length as measured on a large scale map and that, in turn, will be less than the length as measured by a man on foot with a metre rule. As the length of the measuring stick is decreased still further, the size of individual rocks and pebbles becomes important. Eventually the original question becomes practically meaningless as the length becomes practically infinite. ...
...
How then to judge the content of Mandelbrot's essay? It is eccentric but obviously useful. Fractal sets occur in all manner of mathematical models, both old and new, and to have a language in which to discuss them is clearly helpful. Pure Mathematicians may well have felt happy with abstract fractals for many years, but it is obviously healthy if other scientists can also talk freely about them. Mandelbrot has amply made the point that it is helpful to be able to describe some natural objects using the language of fractals rather than the language of smooth continuously differentiable mathematics, at least on some range of scales. But what of Mandelbrot's deeper claims?

To begin with, he makes very few. The introductory section entitled "A manifesto: there is a fractal face to the geometry of Nature" consists largely of the words of others, and includes a long quote from a review by Freeman Dyson of an earlier Mandelbrot Essay. Mandelbrot really only hints; thus we have "Against odds, most of my works turn out to have been the birth pangs of a new scientific discipline" (Foreword), "This bunching of [fractal dimensions] can hardly be a coincidence; it must tell us something profound about the structure of the plane", "This Essay ... promotes a new mathematical and philosophical synthesis." Mathematically, however, the Essay is not satisfying. Mandelbrot is aware that there are problems with his definitions and, reluctantly, opts for the Hausdorff (of Hausdorff-Besicovitch) dimension (1919) as his definition of dimension though it does not produce the result he desires on some sets and is extraordinarily hard to calculate on others. It is probably wise not to have devoted too much of the Essay to these problems, though it is clear that they do need treating if Mandelbrot's ideas are going to be of any real use in the more mathematical areas of possible application (e.g. attracting sets in systems of "chaotic" differential equations).

On the less mathematical side, it is also in the spirit of the Essay that Mandelbrot treats each of his subjects areas in little depth, but he himself says "... I do not consider the fractal point of view as a panacea, and each case analysis should be assessed by the criterion holding in its field, that is, mostly upon the basis of its powers of organisation, explanation, and prediction, and not as an example of mathematical structure." Unfortunately, even for such a shallow treatment, there is relatively little evidence of organisation (beyond description) or explanation in the Essay, and almost no prediction. Some ideas, such as that complicated biological fractals may occur with relatively little genetic coding, have some appeal (to a non-specialist at least); others, such as the spaceship depositing galaxies on a random walk, appeal only to one's more fantastic side. And of the coastline of the British Isles, the reader is likely to learn very little.

Perhaps, now that fractals and fractal-like sets have, with Mandelbrot's help, become a little more familiar, scientists in various fields will begin to come to grips with processes and mechanisms which generate them; in many cases the fractal range of scales lies between the small and large scale regimes already the subject of intensive study. Until then Mandelbrot's models will remain mainly "examples of mathematical structure". And to end, probably unfairly, with some more of Mandelbrot's words, in the meantime "(his) Essay is preface from beginning to end. Any specialist who expects more will be disappointed."

5.1. Review by: Philip E Mirowski.
Journal of Economic Literature 39 (2) (2001), 585-587.

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 self-designated "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 ill-conceived physics envy. In economics, this amounted to repeated exercises arguing that empirical price distributions were fat-tailed, 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 (Multifractals and $1/f$ Noise, 1999).

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". 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 furore by asserting that the Gaussian model was a poor fit, and that the more general Levy-stable family of distributions, derived from a more general limit theorem, gave a better characterisation. 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 characterisations constituted a continuum, with the Gaussian at one extreme, the lognormal an intermediate case, and Levy-stable distributions as the "wild" other extreme. Given the family resemblances, it was deemed unlikely that the question of stochastic characterisation 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, and indeed, has forsworn the goal of a general stochastic characterisation applicable to all markets. ...

5.2. Review by: Bing Hong Wang.
Mathematical Reviews MR1475217 (98i:90019).

This book is a major contribution to the understanding of how speculative prices vary in time with hair-raising swiftness. It presents and tests three successive rules of variation, tackling fast change and long distribution tails, then long dependence in time, and finally both features simultaneously. Careful modelling is needed because prices do not perform a random walk on the street; they are not tossed around like Brownian motion tosses a small piece of matter. Price change is often subjected to substantial sharp discontinuities, and periods of price activity are far from being uniformly spread over time. This book is largely devoted to these topics. One must understand the structure of financial charts thoroughly, including features that fail to bring positive returns to the investor. The understanding gained by a thorough exploration is bound to bring significant knowledge about the mechanisms of the financial markets and about the laws of economics. Most importantly, this knowledge is essential for evaluating the unavoidable risks of trading. The author seeks to establish an ideal model of price variation which can produce sample data streams that are hard to distinguish from actual records, either by eye or by algorithm, and achieves a good part of this goal without ad hoc "patches" or "fixes". Since 1963, the author has presented some new models, including "Levy stable motion", "fractional Brownian motion", "subordination" and "fractional Brownian motion of multifractal time". This book explains all these technical terms and motivates the study of the broad family of processes that they denote. The aim of all these models is to provide more effective ways to handle relatively rare events that have very strong effects. This book incorporates the author's pioneering contributions to finance, but half is entirely new; some chapters are mathematical; others contain few formulas. Here are some highlights: a new classification of forms of randomness into "states" that range from mild to wild; a useful classification of the departure of prices from Brownian motion into Noah and Joseph effects and their combination; a broad panorama of old and new forms of self-affine variability. The tool conceived by the author to tackle discontinuity and concentration is scaling. He concludes that much in economics is self-affine. A good portion of this book studies financial charts as geometric objects with self-affine, fractal scaling by renormalisation. Non-Brownian forms of scaling can be called "non-Fickian". From the viewpoints of economics and finance, the most striking possible consequence of "non-Fickian" scaling is discontinuity/concentration. This book is thoroughly impregnated with the idea of fractals. In fractal technical terms, the author's successive models in finance signify that, once again, the charts generated by his models are self-affine. The study of financial fluctuations has moved on since 1964, and has become increasingly refined mathematically while continuing to rely on Brownian motion and its close kin. Therefore the list of endangered statistical techniques would now include the Markowitz mean-variance portfolios, the Black-Scholes theory and Ito calculus, and the like. In Cootner's words, the author's "view of the economic world is more complicated and much more disturbing than economists have hitherto endorsed". The overall theme is that, while prices vary wildly, scaling rules hold, ensuring that financial charts are examples of fractal shapes. The recognition of the fractal nature of price variation combines many small mysteries into one very large one. This is the first of a multivolume series of Selecta, and cross-references are included.

6.1. Review by: Daniel J M Schertzer and Shaun M Lovejoy.
Mathematical Reviews MR1713511 (2003c:28006).

This book is the second volume of the series "Selected Works of B B Mandelbrot", published by Springer-Verlag (the first was [Fractals and scaling in finance, 1997]). These books are presented as companion books to [The fractal geometry of nature, 1982]. This series has the particularity of being edited by the author himself.

The scientific part of this book mainly corresponds to the re-edition - with occasionally significant editorial changes (in particular in notation) - of papers published by the author between 1962 and 1976 on scaling noises and cascades, including several conference abstracts.

7.1. Review by: Michèle Mastrangelo-Dehen.
Mathematical Reviews MR1878884 (2003a:01026).

This book is a complete and encyclopaedic synthesis of problems and themes in self-affinity, 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.

8.1. Review by: René L Schilling.
The Mathematical Gazette 89 (514) (2005), 168.

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, but the breakthrough came in 1980 with the first picture of the Mandelbrot set in 'Fractal aspects of the iteration of $z \mapsto \lambda z (1 - z)$ for complex $A$ and $z$', Ann. New York Acad. Sci. and reprinted on pp. 37-51 in the present volume. It was, however, Mandelbrot's 1982 masterpiece The Fractal Geometry of Nature that popularised 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 re-read 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 non-specialist) introduction and survey-type 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 lop-sided) personal notes and recollections and always the urge to convey a message.
...
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 hard-to-get journal and proceedings volumes, as it will be appealing to a general informed audience.

8.2. Review by: Martina Zähle.
Mathematical Reviews MR2029233 (2005b:01054).

This is not only the fourth volume of the Selecta - Selected papers of B B Mandelbrot. It is also a philosophically coloured 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 realised his old dream "of helping unscramble the messiness of nature". The present volume emphasises 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 recognised 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.
...
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.

9.1. From the Summary.

Three states of matter - solid, liquid, and gas - have long been known. An analogous distinction between three states of randomness - mild, slow, and wild - arises from the mathematics of fractal geometry. Conventional financial theory assumes that variation of prices can be modelled by random processes that, in effect, follow the simplest 'mild' pattern, as if each uptick or downtick were determined by the toss of a coin. What fractals show, and this book describes, is that by that standard, real prices 'misbehave' very badly. A more accurate, multifractal model of wild price variation paves the way for a new, more reliable type of financial theory.

Understanding fractally wild randomness, also exemplified by such diverse phenomena as turbulent flow, electrical 'flicker' noise, and the track of a stock or bond price, will not bring personal wealth. But the fractal view of the market is alone in facing the high odds of catastrophic price changes. This book presents this view in a highly personal style, with many pictures and no mathematical formulas in the main text.

9.2. Review by: Graham Hoare.
The Mathematical Gazette 92 (523) (2008), 187-188.

For most of his professional life Benoit Mandelbrot has been challenging the orthodoxies of various disciplines. His uncle, Szolem Mandelbrojt, an ardent Bourbakist, advised his nephew to avoid geometry and we know how seriously he took this advice. By the 1970s he had created the subject of fractal geometry which could be exploited to describe and analyse the structural irregularities of the physical world. Almost inevitably, it seems, Mandelbrot was drawn to the world of economics where the volatility and turbulence of markets are well known.

On October 19, 1987, the Dow Jones Industrial Average (DJIA) fell by 508 points, a decrease of 29.2%. Judged by orthodox financial models, the probability of this happening is less than 1 in 1050. Armed with this and other examples, (on this side of the pond, readers will recall the disruption of the European Exchange Rate Mechanism (ERM) in 1992), Mandelbrot proceeds to challenge classical models of modern finance which, since circa 1900, have been underpinned by the Gaussian model of probability. While studying the French exchange, the mathematician Louis Bachelier, using conventional statistical techniques, modelled daily price changes as normally distributed, independent events; but he also acknowledged the existence of outliers. Modern financial models such as Capital Asset Pricing Model (CAPM) and the Black-Scholes Option Pricing Model adopted Gaussian randomness as their foundation. According to the Gaussian model of prices, over the last 90 years, the DJIA should have moved more than 3.4% during a single day only 58 times. In fact it did so on 1001 occasions. It seems that prices are neither normally distributed nor independent. Indeed, the fundamental conclusion of this book is that there is too little skewness and too much kurtosis in financial markets. As an alternative to the classical model, Mandelbrot has developed a multi-fractal model with variable market time, exponential price distributions and fractal generators.

The book is divided into three parts. The first of these, 'The Old Way', deals with early theories of finance and seeks to demonstrate their inadequacy. CAPM, for example, is useful when markets are calm but not when extreme events occur. In the second part, 'The New Way', Mandelbrot explains his multi-fractal approach. However, rather than developing a comprehensive theory of market behaviour, he proposes further investigations. Further, in a paradigm case, his analysis of cotton prices over some 100 years, he uses raw data without seemingly taking into account the underlying changes in production, distribution and the use of such financial instruments as options and futures. It is even more complicated for stocks since markets, regulation and economic situations vary. Whilst Mandelbrot's search for a mathematical model to fit '(mis)behaviour' is laudable, his efforts are weakened by the scant attention he pays to underlying causes. Remove or change these and the theory is vitiated. The third part of the book, 'The Way Ahead', is disappointing. It is a recapitulation of what we have discerned about markets already.

Investors won't get rich after reading this book but they may become more cautious. There are sundry irritations. It is too long and the numerous references to Mandelbrot's genius are tiresome. Many of the diagrams are poor and some fail even to have a scale. The solid mathematical material is assigned to a Notes section at the back. Pace Mandelbrot, most traders agree that the markets move for the most part according to a standard random walk accompanied possibly by a modicum of jump diffusion. This text may well intrigue economists but for mathematicians it is not recommended.

10.1. Review by: Brian Hayes.
American Scientist 101 (1) (2013), 60; 62.

Thirty years ago, when I was new to writing about computation and mathematics, I received a note from Benoit Mandelbrot suggesting we have lunch. I was flattered that the celebrated author of The Fractal Geometry of Nature had noticed my work, and I looked forward to meeting him. When the day came, we had a pleasant meal and a lively conversation. But Mandelbrot had not summoned me merely to eat and chat. He also wanted to lodge a protest: In a recent article I had mentioned the word fractal without also mentioning the name Mandelbrot. I tried to defend myself, pointing out that the word appeared just once, in a passing reference. I noted that I also spoke of gravity without Newton, of geometry without Euclid. But Mandelbrot would have none of this. He looked upon fractal the way Kimberly-Clark looks upon Kleenex - with delight at the term's popularity, but also with dismay that people forget where it came from.

Mandelbrot never gave up his zealous pursuit of proper recognition (He died in October 2010.) His memoir, The Fractalist, continues the campaign. Mandelbrot surely does have claims to fame. Early in his career he did important work on statistical distributions, starting from George Kingsley Zipf's observations about the frequencies of words in written language. The Zipf distribution has "long tails," meaning that rare words are not nearly as rare as one might expect. Mandelbrot showed that long-tailed distributions are themselves not nearly as rare as one might expect; they turn up in many other contexts in addition to linguistics. In the 1960s he demonstrated their particular importance in finance.

Another of his contributions concerned dynamical systems, which trace certain kinds of intricate paths through a mathematical space. Two famous artefacts of this work are the Julia set, which Mandelbrot named after the French mathematician Gaston Julia, and the Mandelbrot set, which others named after Mandelbrot. What about fractals? The word is indisputably Mandelbrot's invention, but beyond that it gets complicated. Much of the underlying mathematics goes back 100 years or more, including the key idea of fractional dimension - which describes objects so wiggly or crumpled that the measure of their size lies somewhere between a line and a plane, or between a plane and a volume. Mandelbrot's contribution to the study of fractals was not to prove deep theorems but to pull together many scattered threads and create what he calls "the first-ever theory of roughness." Then, by force of personality, he persuaded many other mathematicians to pay attention.
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Even when Mandelbrot makes an effort to acknowledge others' work, the gesture somehow gets twisted around and winds up pointing back at Mandelbrot again: "The terms 'Julia set' and 'Lévy process' drew blank stares when I introduced them," he writes. "Today, fractalists use them every day. ... Although some cynics attribute to Julia or Lévy ideas that I originated, I am delighted that this terminology has taken root."

In spite of all the posturing and defensive manoeuvring, Mandelbrot emerges from this story as a surprisingly charming character. I am grateful that he took the time in his last years to set down this testament. Telling the story of one's own life is a human impulse that is too often suppressed, usually on the grounds that "no one would be interested in little old me." Mandelbrot had no trouble on that score.

10.2. Review by: Krzysztof Ciesielski.
Mathematical Reviews MR2975869.

Benoit Mandelbrot (1924-2010) was an outstanding mathematician, widely known as a creator of fractals. His name was famous not only in the mathematical world, so his autobiography would also interest many non-mathematicians. As is written by his wife Aliette in the acknowledgments of the book, Mandelbrot spent years writing his memoir. He died shortly before the manuscript went to the publisher.

In the introduction to the book Mandelbrot writes: "Let me introduce myself. A scientific warrior of sorts, and an old man now, I have written a great deal but never acquired a predictable audience. So, in this memoir, please allow me to tell you who I think I am and how I came to labour for so many years on the first-ever theory of roughness and was rewarded by watching it transform itself into an aspect of a theory of beauty." These words may be regarded as the motivation for writing this book.

Mandelbrot was born in Warsaw to a Lithuanian Jewish family. At 12 years old he moved with his family to Paris. During World War II he studied geometry on his own. In 1945-1947 he was a student of the École Polytechnique (Mandelbrot prefers the name "Carva", used by students and alumni in his time). For the next two years he studied at the California Institute of Technology (he uses the name "Caltech"), then served in the French army for one year and finally obtained his doctorate from the University of Paris in 1952. Then he spent 35 years at IBM Thomas J. Watson Research Center as a research scientist. In the period 1987-2004 he was a professor at Yale University.

Since childhood, Mandelbrot's "Keplerian quest" was very important to him. His childhood dream was to discover something like Kepler did. Kepler used his knowledge of two different fields, i.e., mathematics and astronomy, to notice elliptical orbits of planets and calculate that the "persistent anomalies" existing in the motions of planets were, in fact, not anomalies. This led Mandelbrot to questions like "What shape is a mountain, a coastline, a river, ... a cloud, a flame? How can one describe ... the volatility of prices quoted in financial markets? Could some other number measure the 'overall roughness' of rusted iron, or of broken stone?"

In his memoir Mandelbrot describes the story of his life and achievements. Science and mathematics are mentioned as well, but they are presented in a very popular way, so the book does not require any mathematical knowledge, though awareness of some mathematics will help the reader more deeply understand some remarks in the memoir. In fact, there are no mathematical formulas in the book. The only one allowed by the author appears in the chapter on the Mandelbrot set and is presented as "not having to be understood, appreciated, or acted on" but "it suffices to observe that the formula is very short". It is presented in the words: "Pick a constant $c$ and let the original $z$ be at the origin of the plane; replace $z$ by $z \times z$; add the constant $c$; repeat."

The book is divided into three parts. The first part, "How I came to be a scientist", consists of seven chapters and describes the first 20 years of Mandelbrot's life, up to the end of World War II. Part two, "My long and meandering education in science and in life", contains 13 chapters and ends in 1958. The story presented in the last part begins when Mandelbrot starts working for IBM. In this part, entitled "My life's fruitful third stage", there are 9 chapters.

The memoir is accompanied by a short "Afterword" written by Michael Frame.

In the book Mandelbrot mentions many people he met in his life, including famous mathematicians. In particular, some words are devoted to his uncle Szolem Mandelbrojt, his teachers in Carva: Gaston Julia and Paul Lévy, John von Neumann (Mandelbrot was his last postdoc in Princeton), Norbert Wiener, J. Robert Oppenheimer, Mark Kac, William Feller and his schoolmate in Carva, Valéry Giscard d'Estaing, who later was elected president of France.

In the book there are some pictures and several photographs. Also, several beautiful colour pictures of fractals are included.

Note also the motto of the first part of the memoir, which is a very interesting quotation from Galileo: "All truths are easy to understand once they are discovered; the point is to discover them."

10.3. Review by: Brie Finegold.
Science, New Series 339 (6117) (2013), 274.

Best known for discovering the set named after him, Benoit Mandelbrot's titles ranged from stable boy to IBM scientist, from apprentice toolmaker to eminent mathematician. The autobiography he finished near the end of his life, The Fractalist, puts his accomplishments in historical and mathematical perspective. In summary, he asks, "Does not the distribution of my personal experiences remind one of the central topic of my scientific work - namely, extreme fractal unevenness?" Just as a fractal, his experiences were indeed rough in nature and self-similar in quality.

As a child, Mandelbrot separated from his family and hid his Jewish heritage to survive World War II. Not one to follow advice, he wrote his thesis on an unpopular topic without an advisor. On this note, he quotes mathematician and family friend Jacques Hadamard, apparently complaining about a student who asked for a thesis topic, "Can you imagine that? If he has no topic of his own, he should not even think of a Ph.D.!" Mandelbrot's self described "messy" thesis consisted of two seemingly unrelated parts: one on Zipf's universal power law concerning the frequency of words and one on statistical thermodynamics. Lacking an academic position in Europe, he went to work for IBM after a postdoc at Massachusetts Institute of Technology (MIT). Several universities subsequently invited him for long-term visits: Harvard (in economics), then MIT (in engineering), and finally Yale (in mathematics). But none offered him a full-time position. In 1940s academia, the interdisciplinary nature of his interests was not an asset as it might be today.

Mandelbrot avoided overly technical language, noting that "Many scientific articles are completely flat because they are written for people who do not have to be convinced. ... In my case, the fact that I write for an unknown public influences and shapes my style." Mandelbrot's account will inspire scientists because he writes about the general trajectory of his career and the origins of his thoughts. For example, the study of fractals has its roots in work from the 1910s by Pierre Fatou and Mandelbrot's teacher Gaston Julia. These works lingered in Mandelbrot's mind despite their not being the hot topics of his day. He writes "much of my work has consisted of bringing a medley of old issues back to life and triumphant evolution." In addition, his biggest breakthroughs came later in life. The College de France invited him back at age 49 to talk about his research and subsequently offered him the chance for a position there. Improvements in computer graphics as well as the freedom afforded to him by IBM helped showcase this stunning work.

Needing a title for the slim 1975 work that he calls his "preview book," Mandelbrot coined the term "fractal" based on its similarity to "fractured." A fractal can be built by iterating one transformation of the complex plane and separating points according to characteristics of their orbit under this iteration. Despite the simple algorithms that generate them, fractals have unexpected properties, such as a noninteger Hausdorff dimension and growth that follows power laws. As his work took off, aided by his classic The Fractal Geometry of Nature (2), Mandelbrot saw his disorganisation as his greatness weakness, claiming that it led him to publish work in journals that may not have been the most esteemed or appropriate. "Each partial success aroused some old expectation or some old hunger." Never feeling truly satisfied that his body of work was well presented or complete, he left many open doors for research to continue.

In the end, reliance on instincts for both basic and professional survival paid off. Learning trades and fighting for survival in his youth influenced him to focus on concrete geometry rather than abstract numerics. By assessing pictures qualitatively, he identified opportunities to collaborate where others detected no connection. Teaching, as he confessed, without a clear plan might have enriched his students' lives. He quotes one student from his "Topics in Applied Mathematics" course at Harvard: "I had been told that science was created by humans, but in all my other courses it seemed created by creaky machines. Your course made me watch science being created." Perhaps more students would have witnessed science in action had Mandelbrot been an academic.

Readers might balk at the author's tendency to tell them how to feel through headings such as "A Flawed Ph.D. Dissertation Well Ahead of Its Time" or "Firebrand Newcomer to Finance Advances a Revolutionary Development." But reading The Fractalist, one finds a man who is as self-critical as he is self-promoting. Also, the accounts of his collaborations with Noam Chomsky, Jean Piaget, "Johnny" von Neumann, Stephen Jay Gould, Robert Oppenheimer, and other visionaries indicate that Mandelbrot had little reason to be humble.