How Benoit Mandelbrot’s outsider mathematics turned the broken, the jagged, and the cracked into a theory of the world
A bug on a line printer
The image, when it first appeared in the spring of 1980, looked like an accident. A staff researcher at the Thomas J. Watson Research Center in Yorktown Heights, New York, was running a program on an IBM mainframe — a machine the size of a small refrigerator, with a line printer that drew images by striking ASCII characters in graded densities. On the page that emerged was a smudged, bristly shape, a kind of inkblot or insect, with rough lobes and feathery edges. The man who had asked the computer to produce it was Benoit Mandelbrot, a fifty-five-year-old mathematician of Polish birth, French education, and difficult-to-categorize disposition. He looked at the image and saw what no one else, including the IBM staff who had assumed there was a bug in the code, could yet see: that the irregularities were not noise but content, that the form on the page contained within itself, in miniature, exact replicas of its own bristled outline, and that those copies in turn contained still more copies, all the way down, for as far as any computer could ever look. He had found what would come to be called the Mandelbrot set. It would become, in time, the most reproduced figure in the history of mathematics — the unofficial logo of an idea, the fractal, that would dislodge ancient assumptions about how the world is shaped.
It is hard now, decades into the era of chaos theory and complexity science and “emergence” as the catchword of business books and TED talks, to recover the strangeness of what Mandelbrot did. The mathematics of his time was committed, with a fervor that bordered on the theological, to smoothness. Calculus, the discipline’s most powerful tool, depended on smoothness: a curve had to be tame enough to admit a tangent at every point, an instantaneous direction of travel. The exotic shapes that didn’t behave that way — curves so wiggly they had no tangent anywhere, sets so dust-fine they could not be measured by any conventional ruler — were dismissed as “monsters,” “pathologies,” or, in the chillier phrase of one French analyst, “a gallery of mathematical horrors.” They were exhibits in a freak show. Mandelbrot’s heresy, the one for which he was alternately ignored, mocked, and finally (and grudgingly) accepted, was to argue that the monsters were not exceptions. They were the rule. The smooth curves of the textbooks were what was anomalous, an idealization adopted for the convenience of calculation. The real world — the coastlines, the lungs, the lightning, the leaves, the price of cotton — was rough.
The mathematician without a country
Mandelbrot’s biography reads like a parable about the costs and freedoms of not belonging. Born in Warsaw in 1924 to a Jewish family of cloth merchants and intellectuals, he fled with his parents to France in 1936 as Nazi influence grew, and spent the war years moving among small towns in Vichy France, where he hid, hunted for food, and tutored himself in geometry from old textbooks. Several of his relatives did not survive the war. The episode marked him in ways that lasted. He developed what he called an “eye geometry,” a habit of solving algebraic problems by translating them into shapes; faced with a polynomial, he would picture a curve. This was, in the mathematical culture of the day, an act of mild perversion. The reigning style in France was the abstract, axiom-driven approach of the Bourbaki collective — a group of French mathematicians who, beginning in the 1930s, set out to rebuild the discipline on rigorous, geometry-free foundations. To Bourbaki, intuition was a contaminant. Pictures lied. Mandelbrot, whose uncle Szolem Mandelbrojt was a senior member of Bourbaki and who therefore had a familial stake in the orthodoxy, rebelled.
He bounced. He took his doctorate in Paris, did postdoctoral work at the Institute for Advanced Study in Princeton (where he met John von Neumann, one of the few establishment figures who took to him), passed through MIT, briefly held a position at the University of Lille. None of these stuck. Mandelbrot worked in too many fields — economics, linguistics, fluid dynamics, information theory — and produced results that bore an unsettling family resemblance across them. He had the trespasser’s energy and the trespasser’s penalty: nobody quite wanted to claim him. In 1958, the year his uncle finally helped him secure a tenured French chair, he turned it down. He had been offered a research position at IBM in upstate New York. He took it. He would stay for thirty-five years.
IBM in the late 1950s and 1960s was the closest thing the corporate world had ever produced to a Renaissance court. The company, immensely profitable and mildly paranoid about being seen as a mere supplier of business machines, was willing to pay smart people to think about almost anything, provided they shared occasional results with the engineers. It became the preserve of a generation of misfit scholars who would have been impossible to place in a university. For Mandelbrot, who needed both the freedom to roam and the computational power to render his intuitions visible, IBM was the rare institution prepared to subsidize his trespasses. It would also, not incidentally, give him access to the machines that finally allowed his “monsters” to be seen — not as descriptions in journal articles, but as images on a page.
Monsters in the basement of mathematics
The monsters were old. They had been crawling out of the basement of mathematics since the late nineteenth century, when a series of analysts began constructing objects that were intended, more or less explicitly, as provocations. In 1872, the German mathematician Karl Weierstrass exhibited a function that was continuous everywhere but differentiable nowhere — a curve, in other words, with no smooth direction at any point along it. Henri Poincaré called such functions an outrage against common sense. A few years later, Georg Cantor, the inventor of set theory, described what is now called the Cantor set: take a line segment, remove its middle third, then remove the middle thirds of what remains, and keep going. What is left, after infinitely many removals, is a dust — uncountably many points, but with total length zero. Cantor’s contemporaries were appalled. In 1904, the Swedish mathematician Helge von Koch constructed a curve, the Koch snowflake, by taking an equilateral triangle and adding a smaller triangle to the middle of each side, and then doing the same to each new side, forever. The result was a closed shape with finite area whose perimeter, properly measured, was infinite. In 1915, the Pole Wacław Sierpiński described a triangle of triangles, riddled with holes at every scale, and a carpet made the same way.
These constructions, and a handful of related ones, formed the underworld of mathematics for the first half of the twentieth century. They were generally treated as cautionary tales — exhibits in the case for axiomatic rigor over geometric intuition. They were also studied by a small number of specialists, including a pair of French mathematicians, Pierre Fatou and Gaston Julia, who in the years around the First World War investigated what happens when you take a simple equation — say, take a complex number, square it, add a constant, repeat — and follow the result around the complex plane. They found that, depending on the starting number, the iteration either spiraled off to infinity or settled into some bounded path, and that the boundary between these two fates was a curve of nightmarish intricacy: a “Julia set,” named for the more famous of the two. Julia, who had lost much of his face in the trenches at the Battle of Verdun and lectured for the rest of his life behind a leather mask, won a prize for his work in 1918. Then the field went quiet. Without computers, the Julia sets could be described but not drawn. They went into the basement to wait.
Mandelbrot’s particular insight — the move that, in retrospect, looks both obvious and revolutionary — was to see that all these monsters were members of the same family, and that the family had a name. In a 1975 monograph published in French, expanded in 1977, and revised again in 1982 as The Fractal Geometry of Nature, he coined the term “fractal,” from the Latin fractus, meaning broken or fractured. He liked, he said, that the same Latin root gave us both fracture and fraction — a hint that the new objects would live, dimensionally, between the integers. A fractal, in his definition, was a shape that exhibited some form of self-similarity across scales: zoom in on a piece of it, and you would find, if not an exact replica, then a recognizable cousin of the whole. A coastline. A lung. A snowflake. A lightning bolt. The branching of a tree, the branching of a river, the branching of a neuron. Some were strictly self-similar, like Koch’s snowflake. Most were “statistically self-similar,” which is to say their roughness at one scale resembled, on average, their roughness at another. What unified them was that they had a fractional dimension — a number, between the familiar one and two and three, that measured their roughness.
How long is the coast of Britain?
The clearest path into this last idea, dimension, runs through a small, almost comic question that the British meteorologist Lewis Fry Richardson had asked in the 1950s, near the end of his life. Richardson was a Quaker pacifist who had spent decades trying to find a mathematics of why countries went to war. He had noticed, looking at empirical data, that the probability that two nations would come to blows seemed to depend on the length of their common border. The trouble was that no one could agree on how long the borders were. The figures for the Spanish-Portuguese frontier, for instance, were quoted in different sources at 987 kilometers in one and 1,214 in another — a discrepancy of more than twenty percent. Richardson, in a posthumously published paper, asked the broader version of the question: How long is the coast of Britain?
The answer, he showed, depended on the size of your ruler. If you measured the coastline with a hundred-kilometer yardstick, walking from headland to headland in great strides, you got one number. If you used a ten-kilometer ruler, you began to pick up the bays and the major peninsulas, and you got a longer number. With a one-kilometer ruler, you picked up the coves; with a hundred-meter ruler, the inlets between rocks; with a meter, the rocks themselves; with a centimeter, the grains. The coastline, in other words, did not have a length. It had a length that depended on how you looked at it, and that grew without bound as you looked closer. Richardson observed something else, more subtle: for different coastlines, the rate at which the measured length increased was different. The west coast of Britain, with its fjords and crags, grew faster than the smoother coast of South Africa. There was a number, an exponent, associated with each coast, that quantified its raggedness.
Mandelbrot, in a 1967 paper in the journal Science titled “How Long Is the Coast of Britain?”, took up Richardson’s exponent and gave it a name: a fractal dimension. The coastline of Britain, he calculated, had a dimension of roughly 1.25 — more than a line, less than a plane. The coast of South Africa came in around 1.02, almost smooth. A more jagged coast might be 1.4. The number was a measure of how aggressively the shape filled the space it lived in; it was, in some sense, the geometry of roughness rendered as a single digit. Mandelbrot would spend the next decade extending the idea. The bronchial tree of the human lung, he and others showed, has a fractal dimension approaching three — because the lung is doing everything it can to pack as much surface area as possible into the volume of the chest. The boundary of the Mandelbrot set, his eponymous object, has a fractal dimension of exactly two: it is so wiggly that it essentially fills a plane. The body’s blood vessels, the bark of trees, the surface of a cauliflower, the network of cracks in a dry lake bed: all had dimensions of their own, and the dimensions, taken together, formed a kind of vocabulary for describing the shape of what is.
The pictures that made the case
The most powerful argument for the fractal idea was visual. Beginning in the late 1970s, Mandelbrot and his collaborators at IBM — among them Richard Voss, a young physicist who became a virtuoso of fractal computer graphics — began generating images that demonstrated, at a glance, that the new geometry was the right one for the natural world. They produced landscapes — mountain ranges with rocky ridges and braided valleys — by iterating a few simple rules, and they were indistinguishable from photographs of the Sierras. They drew clouds, planets, river deltas. Star Trek II: The Wrath of Khan, released in 1982, used fractal algorithms, derived from Mandelbrot’s work, to render the “Genesis planet” sequence in what is now regarded as one of the first uses of procedurally generated computer graphics in a major film; Return of the Jedi followed the next year, using the same techniques for the moons of Endor. In August 1985, Scientific American gave the Mandelbrot set a richly colored cover, accompanied by an A.K. Dewdney column on how to draw it on a home computer. The set itself, with its central cardioid and its budded circles and its filamentary tendrils, was suddenly everywhere — on book jackets, on dorm-room posters, in the title sequences of public-television documentaries. It was a peculiar moment in the cultural history of mathematics. A geometric object had become, briefly, a kind of poster — not for any movement or product but, somehow, for the idea that the world was more interesting than calculus had given it credit for being.
Behind the imagery, the science was real, and it spread quickly. Biologists found that the dendritic arbors of nerve cells were fractal, that the surface area of the small intestine was fractal, that the spreading of tumors followed fractal geometries. Geologists found fractal patterns in earthquake distributions, in fault surfaces, in the distribution of oil deposits. Cosmologists noticed that the large-scale distribution of galaxies in the universe seemed, at least up to some cutoff, to exhibit a fractal structure: clusters of galaxies grouped into superclusters, the superclusters threaded by filaments, the filaments forming a kind of cosmic foam. Music theorists analyzed the temporal structure of Bach and Beethoven and found, in some cases, statistically self-similar patterns in the spacing of notes. The branching of a fern, photographed and compared to a recursively generated image, looked the same. The likeness was not always perfect, and it almost never extended over more than a few orders of magnitude — a cauliflower is fractal for three or four levels of zoom, not forever — but the fact that the same mathematics described so many disparate things was, depending on one’s temperament, exhilarating or suspect.
For Mandelbrot himself, the goal was never to suggest that the world is fractal “all the way down.” He was more careful than his interpreters. What he insisted on was the more modest, but still revolutionary, claim that traditional Euclidean geometry — the geometry of cones, spheres, cylinders, lines — was a poor first approximation of natural form, and that fractal geometry was a better one. In a famous passage at the beginning of The Fractal Geometry of Nature, he wrote that clouds are not spheres, mountains are not cones, coastlines are not circles, bark is not smooth, and lightning does not travel in a straight line. The list reads as a kind of inventory of everything that two thousand years of geometric idealization had quietly excluded.
Fat tails and the war with the bell curve
If fractals had remained a doctrine about lungs and ferns, they might have entered the curriculum as a useful corrective and stayed there. Instead, Mandelbrot took them, with characteristic recklessness, into the most heavily defended preserves of twentieth-century intellectual life. He took them into finance.
His route into the subject was, again, biographical. In the early 1960s, while at IBM, Mandelbrot was invited to give a talk at the economics department at Harvard and arrived in the office of an economist named Hendrik Houthakker. On Houthakker’s blackboard was a diagram that Mandelbrot recognized: a humped distribution, with a long horizontal axis and a thin, fat-bellied curve. Mandelbrot, who had been working on the statistics of income distribution, asked why his host had drawn his unpublished data on the board. Houthakker laughed. The diagram had nothing to do with income. It was eight years of daily price changes for cotton.
The coincidence — the same fat-bellied, long-tailed shape appearing in two unrelated economic phenomena — set Mandelbrot off. He requested the cotton-price data, took it back to IBM, and ran it through the company’s mainframe. What he found was not what the textbooks said he should find. The dominant model of financial markets at the time, derived from a 1900 doctoral thesis by the French mathematician Louis Bachelier, held that price changes were the cumulative effect of many small, independent influences and should therefore — by the central limit theorem — be distributed in a bell curve, the familiar Gaussian. Bachelier’s model was the foundation of what became, after the Second World War, modern finance: it underwrote the efficient market hypothesis, the Black-Scholes options-pricing formula, the entire apparatus of quantitative risk management.
The cotton data, Mandelbrot showed, did not obey the bell curve. Its tails were too fat. Days of enormous price change — moves of five or six standard deviations from the mean, events that should, on the Gaussian model, occur once every few millennia — happened with disturbing regularity. The distribution was a member of a different family, what statisticians call a “stable Paretian” or Lévy distribution, with the troubling property that, for sufficiently wild members of the family, the variance is infinite. There was, in effect, no average size for a cotton-market shock. The next move might always be a great deal larger than the last. Mandelbrot also noticed something else, perhaps even more disturbing for the orthodoxy: the price changes were not independent. Volatility came in clusters — calm followed by storm, storm by storm — and the structure of these clusters was, at every timescale he could check, statistically self-similar. The market, in other words, was a fractal.
He published this work in the early 1960s, in journals read by economists. The reception was hostile. The model was inconvenient: it did not lend itself, the way the Gaussian did, to clean closed-form solutions; it implied that risk could not be easily measured because the parameters that would normally measure it did not exist; and it threatened the careers of an emerging generation of financial economists who would, in the coming decades, win Nobel Prizes for work that depended on Bachelier’s framework. The economist Paul Cootner, reviewing Mandelbrot’s findings in his 1964 anthology The Random Character of Stock Market Prices, conceded, in effect, that if Mandelbrot was right, almost the entire toolkit of statistical finance was obsolete. He then declined, with the polite firmness of a doorman, to entertain the possibility. Mainstream finance moved on. Mandelbrot, finding the door closed, went back to his other subjects.
He never fully gave up the argument. In a 2004 book titled The (Mis)behavior of Markets, written with the financial journalist Richard Hudson, he made his case again, this time for a popular audience. The book described, with bone-dry humor, decades of being right and being ignored. Three years later, in the summer of 2007, the credit markets began to seize up; the next year, the financial system convulsed. In the autumn of 2008, hedge-fund managers were quoted in the press explaining that their losses were the result of “twenty-five-standard-deviation events” happening on consecutive days — a statistical formulation that, on the Gaussian model, was approximately as likely as winning the lottery on every day of one’s life. The standard models had not seen the crash coming because the models could not, in principle, see crashes. Some of them, in a coincidence rich enough to delight a satirist, had been engineered by intellectual heirs of the same economists who had dismissed Mandelbrot’s cotton paper forty years before. He himself, then in his mid-eighties, lived long enough to see the vindication and to point it out, with what observers described as a chastened sort of triumph. He died in 2010.
Where the idea runs out
It is tempting, given the shape of the story so far, to render Mandelbrot as a kind of prophet — the outsider who saw the truth before the orthodoxy did, who was punished for his prescience, who lived to be proved right. He himself was not above this self-presentation. His memoir, published posthumously in 2012, was titled The Fractalist, and the subtitle promised a “memoir of a scientific maverick.” But to leave the story there is to skip the more interesting question, which is what the fractal revolution actually accomplished, and where it ran out of road.
The harshest critics of fractal geometry have argued, with some justice, that its claims of universality have been overstated. A coastline is not, strictly, self-similar; it is only statistically self-similar, and only over a limited range of scales. Below the size of a grain of sand and above the radius of a small continent, the model breaks down. A cauliflower is fractal for perhaps four or five generations of zoom; after that, you are looking at cellulose fibers, not florets. Worse, the practice of “measuring the fractal dimension” of natural objects — coastlines, lungs, blood vessels — is, in many empirical implementations, statistically suspect. Different methods give different numbers; the published values vary widely; and in some celebrated cases, careful reanalysis has suggested that there was no fractal scaling there at all. The chemist David Avnir and colleagues, in a much-discussed 1998 article in Science, surveyed the published claims of physical fractality in the literature and concluded that most of them were drawn from scaling ranges of less than two orders of magnitude — too narrow, by any rigorous standard, to qualify. Their wry summary was that fractals appeared to be everywhere and nowhere at once.
The point is fair, and it is worth taking seriously. The fractal idea, like any idea that promises a new key to a wide door, attracted in its train a great deal of loose application — papers in which someone had counted the number of branches in a tree at three different scales and announced a fractal dimension on insufficient evidence. There is a long, melancholy history of this kind of intellectual inflation. The same fate, on a smaller scale, has befallen “complexity,” “emergence,” and “the science of networks.”
And yet. To dismiss fractal geometry on these grounds is to confuse the misuse of an idea with the idea itself. The deep claim of Mandelbrot’s work is not that nature is literally self-similar at all scales. It is that the rough, the broken, and the irregular are not deviations from form but the substance of form. The world is shaped by the way it grows, and growth is recursive: the small repeats the large because the same processes — accretion, branching, erosion, percolation, feedback — operate at every level. To call something a fractal is, in many cases, less a metaphysical claim than a useful piece of bookkeeping: a way of saying that the right description of this thing uses the same vocabulary at all the scales at which it matters. Mandelbrot did not give us a theory of everything. He gave us a vocabulary for the parts of the world that the older theories had been forced to ignore.
Learning to see in fractals
What does it mean to see the world fractally? Most concretely, it means accepting that some things are not measurable in the way we have been trained to measure them. A coastline has no length. A lung has no surface area — or rather, its surface area is a function of the resolution at which you ask. A stock market has no average return — or rather, the average is a fragile statistical fiction that conceals the moves that will determine your fortune. The lesson is not nihilism. It is not that nothing can be measured. It is that the numbers we get out depend on the questions we put in. To know how long a coast is, one must say: at what scale?
In a deeper sense, the fractal way of seeing is a kind of correction to a long inheritance, going back at least to Plato, in which the smooth and the simple were taken as the privileged forms of reality and the rough and the complex as their corruptions. The cone, the sphere, the line — these were the bones of the cosmos, and the things we saw were imperfect copies. Mandelbrot’s work, more than most twentieth-century mathematics, can be read as the reversal of that order. The rough is not a corruption of the smooth. The smooth is an abstraction from the rough. The world began jagged; we have only ever been smoothing it out, in our textbooks, for the sake of being able to compute.
The reversal has political implications, though Mandelbrot rarely drew them. The dominant intellectual style of the modern era — in economics, in management theory, in the social sciences, in policymaking — has been to assume Gaussian distributions, to manage by averages, to plan for the typical and to treat the catastrophic as a footnote. The result has been a long string of catastrophes — financial, ecological, epidemiological — that the averages did not see coming and could not have seen coming. Mandelbrot’s friend and admirer Nassim Nicholas Taleb, the philosopher of risk who popularized the term “black swan,” has spent much of his career as a kind of evangelist for the older man’s quieter heresy: that we live in a world where the unusual is, statistically speaking, where most of the action is, and that any institution which has not internalized this fact is fragile in ways its accountants cannot see. The fractal lesson is that the tail is not a footnote. The tail, in the relevant sense, is the body.
There are also gentler implications. To learn to see fractally is to recover, in some measure, the kind of attention that the world rewards. The shape of a fern; the dendritic spread of a river system seen from the window of a descending airplane; the cracks in old plaster; the whorls of a galaxy and the whorls of a hurricane and the whorls of cream in a cup of coffee: these things, which used to be merely the picturesque, have come to look, under the influence of Mandelbrot’s geometry, like specimens of a single grammar. The world, in this view, is not a museum of distinct objects but a system of related rhymes. It is the kind of insight that resembles, in its texture, what poets and painters have always claimed to know — that the small repeats the large, that everything is connected by family resemblance — but stated in a form that can be quantified and tested. It is a romanticism with numbers.
What survives
Mandelbrot died in October 2010, in a hospice in Cambridge, Massachusetts, of pancreatic cancer. He had been working, in his last months, on extensions of his market models and on his memoir. By the time of his death he had been awarded the Wolf Prize in physics and the Japan Prize, made an officer of the French Legion of Honor, and given an emeritus chair at Yale, where he had finally accepted a university position late in life. The discipline of mathematics, which had spent decades regarding him with the mixture of envy and suspicion typically reserved for trespassers who turn out to have been right, gave him a respectful obituary. There are now international conferences on fractal geometry, courses on fractal markets in business schools, software libraries that allow undergraduates to generate Mandelbrot sets on their laptops in milliseconds — the same images that, in 1980, took the most powerful computer at IBM the better part of an afternoon to produce.
What survives, beyond the images, is the underlying suggestion: that the world is rougher than we have allowed ourselves to admit, that the categories with which we have been measuring it have always been a kind of useful lie, and that a different style of looking — patient, attentive to the small, willing to find the same shape recurring at scales nobody has thought to compare — yields a different and more accurate world. Whether one calls this style a theory of everything, as some of Mandelbrot’s more enthusiastic disciples have done, or a useful corrective, as his more careful successors have argued, is perhaps a matter of temperament. He himself preferred the latter formulation, in his careful moments, and the former when he wasn’t being careful. He had been, after all, an outsider — and outsiders, when they finally get hold of an idea large enough to fight with, are not, on the whole, inclined to undersell it.
The Mandelbrot set is still there, on every computer that cares to draw it. You can zoom in on it forever. Each level reveals something new, but each new thing also looks, eerily, like something you have seen before. There is, after all, a logo for the universe Mandelbrot left us, and this is what it looks like: a black bug, infinitely indented, dreaming smaller and smaller copies of itself, all the way down.
Sources and Further Reading
By Mandelbrot
Mandelbrot, Benoit B. Les objets fractals: forme, hasard et dimension. Paris: Flammarion, 1975. — The first book-length statement of the fractal idea, in French.
Mandelbrot, Benoit B. Fractals: Form, Chance, and Dimension. San Francisco: W. H. Freeman, 1977. — The first English edition.
Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W. H. Freeman, 1982. — The expanded, definitive statement, and the source of the famous opening passage (“Clouds are not spheres, mountains are not cones…”).
Mandelbrot, Benoit B., and Richard L. Hudson. The (Mis)Behavior of Markets: A Fractal View of Financial Turbulence. New York: Basic Books, 2004. — A popular restatement of Mandelbrot’s case against the Gaussian foundations of finance.
Mandelbrot, Benoit B. The Fractalist: Memoir of a Scientific Maverick. New York: Pantheon, 2012. — Posthumous memoir; the primary source for the Warsaw, Vichy, and IBM episodes.
Key papers
Mandelbrot, Benoit B. “The Variation of Certain Speculative Prices.” Journal of Business 36, no. 4 (October 1963): 394–419. — The cotton-price paper.
Mandelbrot, Benoit B. “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension.” Science 156, no. 3775 (May 5, 1967): 636–638.
Richardson, Lewis Fry. “The Problem of Contiguity: An Appendix to Statistics of Deadly Quarrels.” General Systems Yearbook 6 (1961): 139–187. — Posthumous; contains the coastline analysis that Mandelbrot built upon.
Bachelier, Louis. “Théorie de la spéculation.” Doctoral thesis, Sorbonne, 1900. — The Gaussian foundation of modern finance that Mandelbrot spent his career arguing against.
Cootner, Paul H., ed. The Random Character of Stock Market Prices. Cambridge, MA: MIT Press, 1964. — Contains the chapter in which Cootner concedes that, if Mandelbrot is right, conventional statistical finance is obsolete — and then declines to follow the conclusion.
Avnir, David, Ofer Biham, Daniel Lidar, and Ofer Malcai. “Is the Geometry of Nature Fractal?” Science 279, no. 5347 (January 2, 1998): 39–40. — The most cited skeptical survey of fractal claims in the natural sciences.
Historical and biographical context
Gleick, James. Chaos: Making a New Science. New York: Viking, 1987. — The chapter “A Geometry of Nature” remains the best journalistic account of Mandelbrot’s career through the mid-1980s.
Peitgen, Heinz-Otto, and Peter H. Richter. The Beauty of Fractals: Images of Complex Dynamical Systems. Berlin: Springer-Verlag, 1986. — The book that first brought the high-resolution Mandelbrot and Julia set images to a wide audience.
Dewdney, A. K. “Computer Recreations: A Computer Microscope Zooms In for a Look at the Most Complex Object in Mathematics.” Scientific American 253, no. 2 (August 1985): 16–25. — The column, with its color cover, that introduced the Mandelbrot set to home-computer hobbyists.
Hoffman, Jascha. “Benoît Mandelbrot, Mathematician, Dies at 85.” The New York Times, October 16, 2010.
Schroeder, Manfred. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, 1991. — A useful technical companion that places Mandelbrot’s work within the wider mathematics of scaling.
On finance, risk, and the fractal afterlife
Taleb, Nassim Nicholas. Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets. New York: Texere, 2001.
Taleb, Nassim Nicholas. The Black Swan: The Impact of the Highly Improbable. New York: Random House, 2007. — Taleb’s most influential statement of the Mandelbrotian case against Gaussian risk management; written in part as a tribute to Mandelbrot, whom he considered a mentor.
Lo, Andrew W., and A. Craig MacKinlay. A Non-Random Walk Down Wall Street. Princeton, NJ: Princeton University Press, 1999. — A more measured academic engagement with the question of whether market returns satisfy the assumptions of classical finance.
On the precursors
Edgar, Gerald A. Classics on Fractals. Boulder, CO: Westview Press, 1993. — An anthology of the foundational nineteenth- and early-twentieth-century papers by Cantor, Weierstrass, Koch, Sierpiński, Hausdorff, Julia, and Fatou that Mandelbrot pulled together into a single tradition.
My World of Interiors 