mandelbrot fractal

The Misbehavior of Markets: A Fractal View of Financial Turbulence

by Benoit Mandelbrot and Richard L. Hudson  · 7 Mar 2006  · 364pp  · 101,286 words

Aux Armes! Notes Bibliography Index Copyright Page ALSO BY BENOIT B. MANDELBROT Les objets fractals: forme, hasard et dimension (1975, 1984, 1989, 1995) Fractals: Form, Chance and Dimension (1977) The Fractal Geometry of Nature (1982) Fractals and Scaling in Finance: Discontinuity, Concentration, Risk (1997) Fractales, hasard et finance (1959–1997) (1997) Multifracals and 1/f Noise

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: Wild Self-Affinity in Physics (1999) Gaussian Self-Affinity and Fractals: Globality, the Earth, 1/f, and R/S (2002) Fractals, Graphics, and Mathematics Education (With M. L. Frame) (2002) Fractals and Chaos: The Mandelbrot Set and Beyond (2004) TO THE SCIENTIFIC READER: AN ABSTRACT Three states of matter—solid

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they are acknowledged with gratitude. Survival when taking high risks is often a reward for good timing. This is how Professor Mandelbrot repeatedly escaped ruin on his way to fractals. He is deeply in debt to the Thomas J. Watson Research Center of IBM—for thirty-five years a unique haven for

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best speaker of the day—tied only by Steve Ballmer, the Microsoft CEO. As a scientist, Mandelbrot’s fame rests on his founding of fractal geometry, and on his showing how it applies in many fields. A fractal, a term he coined from the Latin for “broken,” is a geometric shape that can

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a stock index or exchange rate. As he puts it, “Roughness is the uncontrolled element in life.” Studying roughness, Mandelbrot found fractal order where others had only seen troublesome disorder. His manifesto, The Fractal Geometry of Nature, appeared in 1982 and became a scientific bestseller. Soon, T-shirts and posters of his most

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famous fractal creation, the bulbous but infinitely complicated Mandelbrot Set, were being made by the thousands. His ideas were also embraced immediately by

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another scientific movement, chaos theory. “Fractals” and “chaos” entered the popular vocabulary. In 1993, on receiving the prestigious

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Wolf Prize for Physics, Mandelbrot was cited for “having changed our view of nature.” MANDELBROT’S LIFE story has been a tale of roughness, irregularity,. and what he

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to combine the formal and the visual. The ready intuition of fractal pictures has, today, made the subject a college course at Yale and other universities, and a popular addition to many high school math courses. But among “pure” mathematicians, Mandelbrot’s approach was initially criticized. Not rigorous, they chided; the eye

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scientific activities and reputation went far beyond the confines of the lab at Yorktown Heights. FOR MANDELBROT, economics has been both inspiration and curse. His study of financial charts in the 1960s helped stimulate his subsequent fractal theories in the 1970s and 1980s. He taught economics for a year at Harvard; and

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. At that time, the theory was beginning to be entrenched in university economics departments—and it would soon become orthodoxy on Wall Street. As Mandelbrot continued his fractal studies, he often returned to economics. Each time, he probed how markets work, how to develop a good economic model for them—and, ultimately

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. It descends partly from a truly extraordinary Web site at http://classes.yale.edu/fractals/index.html created by Mandelbrot’s Yale colleague, Professor Michael Frame, for their popular undergraduate course on fractals for non-science majors, Math 190. Today, Mandelbrot’s message is more timely than ever, after a turbulent decade of bull

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The Old Way Chorale: The computer “bug” as artist, opus 2. (Overleaf) Computer-generated art from Mandelbrot 1982. This design was created by a “bug” in a software program while I was investigating various fractal forms—and it nicely demonstrates the creative power of chance, in art, finance and life. CHAPTER I

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the chief’s house. Finally, within each dwelling is a household altar. The diagram, from Eglash 1999, shows how the village follows a fractal hierarchy. Chaos and the Mandelbrot set. To conclude this pictorial essay, the last example is perhaps the most famous one: a mathematical chimera that my colleagues named for

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T-shirts, prints, book jackets, and PC screen savers. Readers seeking more precise explanations of it may consult my recent book, Mandelbrot 2004a. The Mandelbrot Set illustrates the profound connections between fractal geometry and chaos theory. It uses a remarkably simple mathematical feedback loop to produce an astonishing variety and complexity of results

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portion of it, as if with a microscope, the pattern does not get simpler as you would normally expect. “Proper fractals” remain equally complicated at every level of magnification. But the Mandelbrot set’s complication increases without any bound. Its perplexing mix of simplicity and complexity has made it a mathematical Everest

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a fruitful new way of managing the world’s money and economy. PART THREE The Way Ahead Pharaoh’s breastplate. (Overleaf.) Cover of Mandelbrot 1999a. Illustration of a fractal structure made of an infinity of circles. It is called the limit set of a Kleinian group—another example of the power of

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with extensive comments. So far, four volumes have appeared: Mandelbrot 1997a, 1999a, cover the topics suggested by their titles. The title of Mandelbrot 2002 is less descriptive therefore the contents of several chapters deserve to be singled out. Chapter H0, an overview of fractals and multifractals, is of wide general interest. Chapter H1

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the publisher insisted on its having a title. 117 “To avoid misunderstanding…” These cartoons were sketched in Mandelbrot 1997a and developed in Mandelbrot 2001c. Chapter VII Studies in Roughness: A Fractal Primer 138 “The curve is crinkly…” Fractal dimension is an intricate topic, as you might expect. There are several variant definitions, each suited

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do this forever, and at each stage get an entirely different picture. Its study has become a classic problem in pure mathematics. The Mandelbrot set belongs to both fractal geometry and chaos theory. A chaotic system, far from being disorganized or non-organized, starts with one particular point and cranks it through

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first tests were reported in Mandelbrot, Calvet, and Fisher 1997, Calvet, Fisher, and Mandelbrot 1997, and Fisher, Calvet, and Mandelbrot 1997. 216 “In fact, this concept…” See Mandelbrot and Taylor 1967. 216 “I co-authored in 1967” In that paper, trading time was not taken to be multifractal, but fractal—but neither term was used

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Alternative Models. Research report available at: http://www.ferf.org. Fisher, Adlai, Laurent Calvet, and Benoit Mandelbrot. 1997. Multifractality of Deutschemark/US dollar exchange rates. Cowles Foundation Discussion Paper 1166. Frame, Michael and Benoit B. Mandelbrot. 2002. Fractals, Graphics and Mathematics Education. Washington, D.C.: Mathematical Association of America. Gleick, James. 1987. Chaos

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) : 278A ; 289-292 et 355-358. • Reprint : Chapter N16 of Mandelbrot 1999a. Mandelbrot, Benoit B. 1975. Les objets fractals : forme, hasard et dimension. Paris : Flammarion. Mandelbrot, Benoit B. 1982. The Fractal Geometry of Nature. New York: W.H. Freeman & Co. Mandelbrot, Benoit B. 1985. Self-affine fractals and fractal dimension. Physica Scripta 32: 257-260. • Reprint: Dynamics of

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Fractal Surfaces. Edited by Fereydoon Family & Tamas Vicsek. Singapore: World Scientific, 1991, 11

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-20. • Reprint Chapter H21 of Mandelbrot 2002. Mandelbrot, Benoit B

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. 1986. Self-affine fractal sets, I: The basic fractal dimensions, II: Length and

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area measurements, III: Hausdorff dimension anomalies and their implications. Fractals in Physics. Edited by Luciano Pietronero & Erio Tosatti, Amsterdam: North-Holland, 3-28. • Reprint

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of Part I: Dynamics of Fractal Surfaces. Edited by Fereydoon Family & Tamas Vicsek. Singapore: World Scientific, 1991, 21-36. • Reprint in Chapters H22, H23, H24 of Mandelbrot 2002. Mandelbrot, Benoit B. 1990. Limit lognormal multifractal measures. Frontiers of Physics: Landau Memorial Conference (Tel Aviv

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, 1988). Edited by E. A. Gotsman et al. New York: Pergamon, 309-340. Mandelbrot, Benoit B. 1997a. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. New York

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: Springer-Verlag. Mandelbrot, Benoit B. 1997b. Fractales, hasard et finance. Paris: Flammarion. Mandelbrot, Benoit B. 1997c. Three fractal models in finance: Discontinuity, concentration, risk. Economic Notes (Banca Monte dei Paschi

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di Siena SpA) 26 (2): 197-212. Mandelbrot, Benoit B. 1997d. Les fractales et la bourse. Pour la Science 242: 16

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-17. Mandelbrot, Benoit B. 1999a. Multifractals & 1/f Noise: Wild Self-Affinity in Physics. New York: Springer-Verlag. Mandelbrot, Benoit B. 1999b. Renormalization and fixed points in finance, since 1962

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. Physica A 263: 477-487. Mandelbrot, Benoit B. 1999c. A fractal walk down Wall Street. Scientific American February

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: 70-73. Mandelbrot, Benoit B. 1999d. Survey of multifractality in finance. Cowles Foundation

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Discussion Paper 1238. Mandelbrot, Benoit B. 1999e. Randonnées multifractales à Wall Street. Les mathématiques sociales. Paris

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multifractal time. Quantitative Finance 1: 427-440. Mandelbrot, Benoit B. 2001d. Scaling in financial prices, IV: Multifractal concentration. Quantitative Finance 1: 641-649. Mandelbrot, Benoit B. 2001e. Stochastic volatility, power-laws and long memory. Quantitative Finance 1: 558-559. Mandelbrot, Benoit B. 2002. Gaussian Self-Affinity and Fractals: Globality, the Earth, 1/f Noise

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, and R/S. New York: Springer Verlag. Mandelbrot, Benoit B. 2003. Heavy tails in finance for independent or multifractal price increments. Handbook on Heavy

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T. Ziemba): 1, 1-34. • Related paper: Journal of Statistical Physics 110, 2003, 739-777. Mandelbrot, Benoit B. 2004a. Fractals and Chaos: The Mandelbrot Set and Beyond. New York: Springer Verlag. Mandelbrot, Benoit B. 2004b. Updated reprint of Mandelbrot 1997a. Mandelbrot, Benoit B., Adlai Fisher, and Laurent Calvet. 1997. A multifractal model of asset returns. Cowles

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with random curve random dusts in random in roughness described by self-affine self-similar Sierpinski gasket as society symmetry of web as The Fractal Geometry of Nature (Mandelbrot) France Telecom French, Kenneth R. paper by Friedman, Milton Fundamental analysis concept of value in risk understood in FXTrade Galileo GARCH. See Generalized

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cotton mystery solved by eureka moment of Hurst heard of by IBM work of main work of persistence of Mandelbrot fractals Marcus, Alan J. Market behavior Bachelier on deceptiveness of efficiency in five rules of inherent uncertainty in investment bubbles in mathematical study of misleading in

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Resources Research Williams, Albert L. Wind tunnel Wind turbulence World Trade Center attack Zigzag generator fractal geometry with Zipf, George Kingsley formula of power law slope of word frequencies of Zurich Copyright © 2004 by Benoit B. Mandelbrot All rights reserved. No part of this book may be reproduced in any manner whatsoever

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, (800) 255-1514 or e-mail special.markets@perseusbooks.com. Library of Congress Cataloging-in-Publication Data Mandelbrot, Benoit B. The (mis)behavior of markets : a fractal view of risk, ruin, and reward / Benoit B. Mandelbrot and Richard L. Hudson. p. cm. Includes bibliographical references and index. eISBN : 978-0-465-00468-3

Chaos: Making a New Science

by James Gleick  · 18 Oct 2011  · 396pp  · 112,748 words

. Experiment joins theory. From one dimension to many. Images of Chaos The complex plane. Surprise in Newton’s method. The Mandelbrot set: sprouts and tendrils. Art and commerce meet science. Fractal basin boundaries. The chaos game. The Dynamical Systems Collective Santa Cruz and the sixties. The analog computer. Was this science? “A

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at more closely, the bursts, too, contained error-free periods within them. And so on—it was an example of fractal time. At every time scale, from hours to seconds, Mandelbrot discovered that the relationship of errors to clean transmission remained constant. Such dusts, he contended, are indispensable in modeling intermittency. The

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the verb frangere, to break. The resonance of the main English cognates—fracture and fraction—seemed appropriate. Mandelbrot created the word (noun and adjective, English and French) fractal. IN THE MIND’S EYE, a fractal is a way of seeing infinity. Imagine a triangle, each of its sides one foot long. Now imagine

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his exploration of infinitely complex shapes had an intellectual intersection: a quality of self-similarity. Above all, fractal meant self-similar. Self-similarity is symmetry across scale. It implies recursion, pattern inside of pattern. Mandelbrot’s price charts and river charts displayed self-similarity, because not only did they produce detail at

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River. Lamont-Doherty is where Christopher Scholz, a Columbia University professor specializing in the form and structure of the solid earth, first started thinking about fractals. While mathematicians and theoretical physicists disregarded Mandelbrot’s work, Scholz was precisely the kind of pragmatic, working scientist most ready to pick up the tools of

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fractal geometry. He had stumbled across Benoit Mandelbrot’s name in the 1960s, when Mandelbrot was working in economics and Scholz was an M.I.T. graduate student spending a great deal of time on

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seismologists had been content to note the fact and move on. Scholz remembered Mandelbrot’s name, and in 1978 he bought a profusely illustrated, bizarrely erudite, equation-studded book called Fractals: Form, Chance and Dimension. It was as if Mandelbrot had collected in one rambling volume everything he knew or suspected about the

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a few years this book and its expanded and refined replacement, The Fractal Geometry of Nature, had sold more copies than any other book of high mathematics. Its style was abstruse and exasperating, by turns witty, literary, and opaque. Mandelbrot himself called it “a manifesto and a casebook.” Like a few counterparts

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surfaces, and surfaces were everywhere in this book. He found that he could not stop thinking about the promise of Mandelbrot’s ideas. He began to work out a way of using fractals to describe, classify, and measure the pieces of his scientific world. He soon realized that he was not alone

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they become so narrow that blood cells are forced to slide through single file. The nature of their branching is fractal. Their structure resembles one of the monstrous imaginary objects conceived by Mandelbrot’s turn-of–the-century mathematicians. As a matter of physiological necessity, blood vessels must perform a bit of

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of medium-sized arteries. The transitional regions…are often designated arteries of mixed type.” Not immediately, but a decade after Mandelbrot published his physiological speculations, some theoretical biologists began to find fractal organization controlling structures all through the body. The standard “exponential” description of bronchial branching proved to be quite wrong; a

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argued that one key to understanding heartbeat timing was the fractal organization of the His-Purkinje network, a labyrinth of branching pathways organized to be self-similar on smaller and smaller scales. How did nature manage to evolve such complicated architecture? Mandelbrot’s point is that the complications exist only in the

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context of traditional Euclidean geometry. As fractals, branching structures can be described with transparent simplicity, with just a few bits of information. Perhaps the

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for goose down, it was by finally realizing that the phenomenal air-trapping ability of the natural product came from the fractal nodes and branches of down’s key protein, keratin. Mandelbrot glided matter-of-factly from pulmonary and vascular trees to real botanical trees, trees that need to capture sun and

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resist wind, with fractal branches and fractal leaves. And theoretical biologists began to speculate that fractal scaling was not just common but universal in morphogenesis

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Pope, casting his benedictions from one side of the field to the other. They fought back. Scientists could hardly avoid the word fractal, but if they wanted to avoid Mandelbrot’s name they could speak of fractional dimension as Hausdorff-Besicovitch dimension. They also—particularly mathematicians—resented the way he moved in

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play with them; they were as primary as the elements of Euclid. Simple computer programs to draw fractal pictures made the rounds of personal computer hobbyists. Mandelbrot found his most enthusiastic acceptance among applied scientists working with oil or rock or metals, particularly in corporate research centers. By the middle of the

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an indispensable language and a catalogue of surprising pictures of nature. As Mandelbrot himself acknowledged, his program described better than it explained. He could list elements of nature along with their fractal dimensions—seacoasts, river networks, tree bark, galaxies—and scientists could use those numbers to make predictions. But physicists wanted to

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finite space, confined by a box. How could that be? How could infinitely many paths lie in a finite space? In an era before Mandelbrot’s pictures of fractals had flooded the scientific marketplace, the details of constructing such a shape were hard to imagine, and Lorenz acknowledged an “apparent contradiction” in

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patches of red. And so on—the boundary finally revealed to Hubbard a peculiar property that would seem bewildering even to someone familiar with Mandelbrot’s monstrous fractals: no point serves as a boundary between just two colors. Wherever two colors try to come together, the third always inserts itself, with a

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property that every point on it borders all three regions. And, as the insets show, magnified segments reveal a fractal structure, repeating the basic pattern on smaller and smaller scales. THE MANDELBROT SET IS the most complex object in mathematics, its admirers like to say. An eternity would not be enough time

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that hang, infinitely variegated, like grapes on God’s personal vine. Examined in color through the adjustable window of a computer screen, the Mandelbrot set seems more fractal than fractals, so rich is its complication across scales. A cataloguing of the different images within it or a numerical description of the set’s

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was the meaning it had for the mathematicians who slowly understood it. Many fractal shapes can be formed by iterated processes in the complex plane, but there is just one Mandelbrot set. It started appearing, vague and spectral, when Mandelbrot tried to find a way of generalizing about a class of shapes known

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he was done, he wanted to understand it, and indeed, he finally claimed that he did understand it. If the boundary were merely fractal in the sense of Mandelbrot’s turn-of–the-century monsters, then one picture would look more or less like the last. The principle of self-similarity at

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what the electronic microscope would see at the next level of magnification. Instead, each foray deeper into the Mandelbrot set brought new surprises. Mandelbrot started worrying that he had offered too restrictive a definition of fractal; he certainly wanted the word to apply to this new object. The set did prove to contain

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say—what is important? Is it the straight line, or is it the fractal object?” At Cornell, meanwhile, John Hubbard was struggling with the demands of commerce. Hundreds of letters were flowing into the mathematics department to request Mandelbrot set pictures, and he realized he had to create samples and price lists

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shape was, in this sense, the simpler would be the appropriate rules. Barnsley quickly found that he could generate all the now-classic fractals from Mandelbrot’s book. Mandelbrot’s technique had been an infinite succession of construction and refinement. For the Koch snowflake or the Sierpiński gasket, one would remove line segments

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randomly explore their every dynamical possibility…. Exciting variety, richness of choice, a cornucopia of opportunity. John Hubbard, exploring iterated functions and the infinite fractal wildness of the Mandelbrot set, considered chaos a poor name for his work, because it implied randomness. To him, the overriding message was that simple processes in nature

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; the former is perhaps a bit more historically oriented. For anyone interested in the origins of fractal geometry, the indispensable, encyclopedic, exasperating source is Benoit Mandelbrot, The Fractal Geometry of Nature (New York: Freeman, 1977). The Beauty of Fractals, Heinz-Otto Peitgen and Peter H. Richter (Berlin: Springer-Verlag, 1986), delves into many areas

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WOULD BE “Simple Mathematical Models,” p. 467. “THE MATHEMATICAL INTUITION” Ibid. A GEOMETRY OF NATURE A PICTURE OF REALITY Mandelbrot, Gomory, Voss, Barnsley, Richter, Mumford, Hubbard, Shlesinger. The Benoit Mandelbrot bible is The Fractal Geometry of Nature (New York: Freeman, 1977). An interview by Anthony Barcellos appears in Mathematical People, ed. Donald J

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. Albers and G. L. Alexanderson (Boston: Birkhäuser, 1985). Two essays by Mandelbrot that are less well known and extremely interesting are

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“On Fractal Geometry and a Few of the Mathematical Questions It Has Raised,” Proceedings of the Inter national Congress of Mathematicians, 16

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Processes,” Nature 322 (1986), pp. 789–93; Richard Voss, “Random Fractal Forgeries: From Mountains to Music,” in Science and Uncertainty, ed. Sara Nash (London: IBM United Kingdom, 1985). CHARTED ON THE OLDER MAN’S BLACKBOARD Houthakker, Mandelbrot. WASSILY LEONTIEF Quoted in Fractal Geometry, p. 423. INTRODUCED FOR A LECTURE Woods Hole Oceanographic Institute

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, August 1985. BORN IN WARSAW Mandelbrot. BOURBAKI Mandelbrot, Richter. Little has been written about Bourbaki even now; one playful introduction is Paul R

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. 88–89. MATHEMATICS SHOULD BE SOMETHING Smale. THE FIELD DEVELOPS Peitgen. PIONEER-BY–NECESSITY “Second Stage,” p. 5. THIS HIGHLY ABSTRACT Mandelbrot; Fractal Geometry, p. 74; J. M. Berger and Benoit Mandelbrot, “A New Model for the Clustering of Errors on Telephone Circuits,” IBM Journal of Research and Development 7 (1963), pp. 224

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. 1, for example. WONDERING ABOUT COASTLINES Ibid., p. 27. THE PROCESS OF ABSTRACTION Ibid., p. 17. “THE NOTION” Ibid., p. 18. ONE WINTRY AFTERNOON Mandelbrot. THE EIFFEL TOWER Fractal Geometry, p. 131, and “On Fractal Geometry,” p. 1663. 102 ORIGINATED BY MATHEMATICIANS F. Hausdorff and A. S. Besicovich. “THERE WAS A LONG HIATUS

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” Mandelbrot. IN THE NORTHEASTERN Scholz; C. H. Scholz and C. A. Aviles, “The Fractal Geometry of Faults and Faulting,” preprint, Lamont-Doherty Geophysical Observatory; C. H. Scholz, “Scaling Laws for Large Earthquakes,” Bulletin of the

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MORE Leo Kadanoff, for example, asked “Where is the physics of fractals?” in Physics Today, February 1986, p. 6, and then answered the question with a new “multi-fractal” approach in Physics Today, April 1986, p. 17, provoking a typically annoyed response from Mandelbrot, Physics Today, September 1986, p. 11. Kadanoff’s theory

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, Mandelbrot wrote, “fills me with the pride of a father—soon to be a grandfather

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MET Barnsley. RUELLE SHUNTED IT BACK Barnsley. JOHN HUBBARD, AN AMERICAN Hubbard; also Adrien Douady, “Julia Sets and the Mandelbrot Set,” in pp. 161–73. The main text of The Beauty of Fractals also give a mathematical summary of Newton’s method, as well as the other meeting grounds of complex dynamics

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discussed in this chapter. “NOW, FOR EQUATIONS” “Julia Sets and the Mandelbrot Set,” p. 170. HE STILL PRESUMED Hubbard. A BOUNDARY BETWEEN TWO COLORS Hubbard; The Beauty of Fractals; Peter H. Richter and Heinz-Otto Peitgen, “Morphology of Complex Boundaries,” Bunsen-Gesellschaft für Physikalische Chemie 89

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1985), pp. 575–88. THE MANDELBROT SET A readable introduction, with instructions for writing a do-it–yourself microcomputer program

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AN INCREDIBLE VARIETY “Julia Sets and the Mandelbrot Set,” p. 161. IN 1979 MANDELBROT DISCOVERED Mandelbrot, Laff, Hubbard. A first-person account by Mandelbrot is “Fractals and the Rebirth of Iteration Theory,” in The Beauty of Fractals, pp. 151–60. AS HE TRIED CALCULATING Mandelbrot; The Beauty of Fractals. MANDELBROT STARTED WORRYING Mandelbrot. NO TWO PIECES ARE “TOGETHER” Hubbard. “EVERYTHING

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WAS VERY GEOMETRIC” Peitgen. AT CORNELL, MEANWHILE Hubbard. RICHTER HAD COME TO COMPLEX SYSTEMS Richter. “IN A BRAND NEW AREA” Peitgen. “RIGOR IS THE STRENGTH” Peitgen. FRACTAL BASIN BOUNDARIES Yorke; a good introduction

The Fractalist

by Benoit Mandelbrot  · 30 Oct 2012

trademarks of Random House, Inc. Library of Congress Cataloging-in-Publication Data Mandelbrot, Benoit B. The fractalist : memoir of a scientific maverick / Benoit Mandelbrot. p. cm. eISBN: 978-0-307-37860-6 1. Mandelbrot, Benoit B. 2. Mathematicians—France—Biography. 3. Fractals. I. Title. QA29.M34A3 2012 510.92—dc22 [B] 2012017896 www.

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pantheonbooks.com Cover image Benoit Mandelbrot. Emilio Segre Visual Archives/American Institute of Physics/Photo

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the memory of Johannes Kepler, who brought ancient data and ancient toys together and founded science. Contents Cover Title Page Copyright Dedication Acknowledgments by Aliette Mandelbrot Beauty and Roughness: Introduction Part One: How I Came to Be a Scientist 1. Roots: Of Flesh and the Mind 2. Child in Warsaw,

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Addiction to Classical Music, Voice, and Opera 13. Life as a Grad Student and Philips Electronics Employee, 1950–52 14. First Kepler Moment: The Zipf-Mandelbrot Distribution of Word Frequencies, 1951 15. Postdoctoral Grand Tour Begins at MIT, 1953 16. Princeton: John von Neumann’s Last Postdoc, 1953–54 17. Paris

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Moving from Place to Place and Field to Field, 1964–79 25. Annus Mirabilis at Harvard: The Mandelbrot Set and Other Forays into Pure Mathematics, 1979–80 26. A Word and a Book: “Fractal” and The Fractal Geometry of Nature 27. At Yale: Rising to the University’s Highest Rank, Sterling Professorship, 1987

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–2004 28. Has My Work Founded the First-Ever Broad Theory of Roughness? 29. Beauty and Roughness: Full Circle Afterword by Michael Frame Inserts Illustration Credits About the Author Acknowledgments by Aliette Mandelbrot

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of mathematics, my unabashed play with abandoned “pathologies” led me to a number of far-flung discoveries. An exquisitely complex shape now known as the Mandelbrot set has been called the most complex object in mathematics. I pioneered the examination of reams of pictures and extracted from them many abstract conjectures

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, Gaston Julia (1893–1978) and Pierre Fatou (1878–1929). Around 1980, I had the privilege of joining Montel’s scientific progeny by discovering the Mandelbrot set, the work that made our family name known far and wide. It revived sober words and formulas of Montel’s school in the 1910s

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later. In 1979, the physicist Richard P. Feynman (1918–88) invited me to return to Caltech to give a lecture on fractals (this was right before I discovered the Mandelbrot set). He and Delbrück sat next to each other just under my nose. Throughout, Feynman nodded and smiled approvingly. Delbrück remained stone

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École Polytechnique to the general commanding the armed forces in Paris. They knew each other, and the letter said: “Dear Friend. A graduating student, Benoit Mandelbrot, needs an exit visa to take a scholarship in the United States. His military record looks ridiculously complicated. I take it upon myself to inform

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a deliberation that lasted thirty years did I feel up to facing quadratic dynamics, and I discovered something that became its most recognized icon—the Mandelbrot set. Instead, I wrote a somewhat strange two-part dissertation for the Doctorat d’État ès Sciences, which was soon overtaken by far better

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many parents ready for any personal sacrifice for the sake of their children, none had come close to Father. 14 First Kepler Moment: The Zipf-Mandelbrot Distribution of Word Frequencies, 1951 “TAKE THIS REPRINT. That’s the kind of silly stuff only you can like.” These words of Uncle Szolem—

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the graphs, they covered many fields and were fascinating. They contradicted Zipf’s claim about word frequencies that Walsh had accepted—but confirmed the Zipf-Mandelbrot formula-to-be! So I could respond to my friends by broadening Plutarch’s advice: to admire part of a man’s works, you

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not sufficient reason to disregard him. To my herd-averse, rebellious side, it may even have been a plus. In short time, the Zipf-Mandelbrot formula became part of my Ph.D. dissertation. Then other graphs in Zipf’s book filled several years with interesting developments. I then left Zipf

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language as it stood in 1950. That wrinkle—statistical thermodynamics—is one of the most sublime pillars of physics. The key feature of the Zipf-Mandelbrot formula exponent was inherited from the statistical thermodynamics motivation: a “temperature of discourse.” It could measure differences from text to text, from speaker to

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ride witnessed the first of many Kepler moments in my life. Soon after it, I examined Zipf’s book. His charts confirmed that the Zipf-Mandelbrot formula was a vast improvement. A difficulty: a well-defined probability may exist for common words, but what about rare words, especially in multiauthor

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I stood frozen with gaping mouth as the physicist J. Robert Oppenheimer, father of the atom bomb, sprung up. “May I respond, Otto? If Dr. Mandelbrot will allow, I would like to make a few comments. The title listed in the announcement of this lecture was tentative and should have been

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have been shared with the audience. As he sat down, the mathematician John von Neumann, father of the computer, stood up. “I invited Dr. Mandelbrot to spend the year here, and we have had very interesting conversations. If he allows me, I would like to sketch some points that Oppie

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apply thermodynamics to social science problems but fails; you have actually achieved something.” He was especially thrilled to hear that my story of the Zipf-Mandelbrot law of word frequencies involved the notion of temperature of discourse. This fundamental exponent is usually greater than 1, but in certain special cases is

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related topic that he could handle, one that led to new mathematics, yet was distinctly traditional, while Hurst’s was not. After the Hurst-Mandelbrot theory had solved the empirical riddle, I asked Feller to stop by my IBM office during one of his visits. To be frank, I set

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academic power and felt, at that point, that the wave of the future was quantitative history. They held an overly enthusiastic interpretation of the Zipf-Mandelbrot law of linguistics and of my effectiveness in Geneva with the psychologist Jean Piaget. So they invited me to set up an ambitious research group

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counterpart of liquid being “slow” randomness. And liquids happen to be enormously more complicated to study. A Backhanded Compliment? There can be little doubt that Mandelbrot’s hypotheses are the most revolutionary development in the theory of speculative prices since Bachelier’s initial work [of 1900. His] papers force us to

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an integral part of his argument evidence of a more complicated and much more disturbing view of the economic world than economists have hitherto endorsed.… Mandelbrot, like Prime Minister Winston Churchill before him, promises us not utopia, but blood, sweat, toil and tears. If [he] is right, almost all our

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going to join the faculty. There is irony in this. This was the same Fama who, in 1964, submitted a thesis subtitled “A Test of Mandelbrot’s Stable Paretian Hypothesis.” He believed that successive price changes were statistically independent. I had to convince him that I had never claimed independence and

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prided itself on bringing the two of us together. It seems that many big dams are built in China. I wonder whether they are Hurst-Mandelbrots. Distribution of Galaxies That the Milky Way is one of a number of similar “objects” in the sky is a surprisingly recent notion: it

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of continuing education that my overworked local friends could only dream of. The substance of my talks, prepared at IBM, concerned the first (Pareto-Lévy-Mandelbrot) of my three successively improved models of financial prices. The creative aspect involved new input that triggered late work in hydrology and the second (Hölder

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something he had heard about the variability observed in the discharges of rivers. I became very excited. This was about the time when the Berger-Mandelbrot paper on telephone errors was published. Also, economics had led me to worry about oil fields. So I knew that two examples of scaling

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equipment, one examines its performance under multifractal variability. This is even a fairly big business, from what I understand. 25 Annus Mirabilis at Harvard: The Mandelbrot Set and Other Forays into Pure Mathematics, 1979–80 WHAT I ACHIEVED OR STARTED during the spring of 1980 went well beyond the wildest dreams

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pictures were intriguing objects I then called lambda and mu-ma—alternative ways of representing a fundamental new mathematical structure that became known as the Mandelbrot set. It has been called the most complex object in mathematics, has become a topic of folklore, and remains my best and most widely

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a bold conjecture: that solving the usual partial differential equations of physics can yield either familiar and expected smoothness, or fractality. A Grand Old Problem Frozen in Time How did the Mandelbrot set arise and provoke such strong reaction? Basically, from a challenge that I “inherited” from Uncle Szolem when I

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letters and three symbols. In mathematical lingo, this is a quadratic map, something close to an ancient curve called a parabola. But in the Mandelbrot set, z denotes a point in the plane, and the formula expresses how a point’s position at some instant in time defines its position

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the circumference of a circle and the curvature gradually “irons out,” yielding an increasingly straight line. But zoom instead toward a boundary point on the Mandelbrot set and what you see becomes ever more beautiful, wild, baroque, and complex in many distinct ways, which the set of color images in this

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have heard it described as “pretty—yet pretty useless.” Important applications of new discoveries take time to be revealed, and we have seen that the Mandelbrot set has powerful redeeming features. Thought wanders to Napoleon’s saying that a good sketch, in all its complexity, is worth a thousand words, or

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—and would have been thrilled to live with it far longer, were it not that success invites too many other seekers. Preview of the Mandelbrot Set at the New York Academy of Sciences The mathematical theory of chaos was a hot topic in the late 1970s and the focus of

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a major reference book. When the time came to turn in my section, I submitted instead the first announcement of the key facts about the Mandelbrot set. The announcement included several of those early pictures. Worried that the printer would think the pictures were ink smudges, I added this instruction: “

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into something entirely different: the first-ever public discussion of my discovery of the object that—very late that year—came to be called the Mandelbrot set. Peter’s help was essential to that discovery. Ultimately, Harvard did not work out. I was expected to pursue and teach my style

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Foundation was responding to critiques like mine by establishing a supercomputer at Geometry Center at the University of Minnesota. Wide Wonder, Complexity, and Mystery The Mandelbrot set strongly appeals to three very different groups to which I belong: those interested in pictures, complexity, and pure mathematics. Pictures The complication of the

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the complexity of a mathematical structure. They put forward the length of the shortest sentence that could implement that structure. Where does this position the Mandelbrot set? Is it the most complex set in the whole of mathematics, as some have asserted, or is it as simple as its generating

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be restated in a different way. But given the stark discontinuous contrast between an input and an output that today is nearly instantaneous for the Mandelbrot set, many view it as extremely—miraculously!—complex. I feel exceptionally privileged that my wanderer’s life led me to be the agent of

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had grown completely familiar, as though one had always seen them. Incredible! (Illustration Credit 25.7) How does the importance of the Mandelbrot set compare to that of fractal finance, which is highly influential in a well-defined community of “practical people”? All my diverse “children of the mind” are equally

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subfields by suddenly settling many seemingly unrelated conjectures. It was the first Fields in probability theory, but in previous years my key conjecture concerning the Mandelbrot set had already led to two Fields. In an ironic way, my disregard of the customary division of labor has advertised that, in mathematics,

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in life, when I did my best-known work. My refoundation of finance was to occur as I neared forty, and the discovery of the Mandelbrot set came at fifty-five. For a scientist, those are unusually—astonishingly—old ages, as many witnesses have noted. And the number of would-

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a major article appeared in which he listed the greatest designs ever. The list included the Book of Kells and the Taj Mahal … and the Mandelbrot set! That was an extremely strong statement, and I was pleased to meet him shortly afterward. We have had interesting times together, including serious public

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And to remain embattled. How come? Perhaps by fluke, but I think mostly for a reason. Afterword Michael Frame, professor of mathematics, Yale University BENOIT MANDELBROT died shortly before he could make final revisions to this memoir. Aliette, his wife of many years, asked me to write this afterword. I hope

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was based on a simple idea—scaling, iteration, and dimension—applied with great finesse in new settings. By far, the biggest surprise is the Mandelbrot set. In class we set up the simple formula and describe the iteration process and how to color-code the result. Then we run the

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of fractal landscapes Fractal painting of flowers, Augusto Giacometti (Illustration Credit bm1.12) Cast of a human lung (Illustration Credit bm1.13) The Great Wave, Hokusai (Illustration Credit bm1.14) Rough deposit of gold (Illustration Credit bm1.15) Turbulence on Jupiter (Illustration Credit bm1.16) Science, Art, and Nature Zooming into the Mandelbrot

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(Illustration Credit bm1.19) Deep into the Mandelbrot set (Illustration Credit bm1.20) “Cave painting,” modified Mandelbrot set fragment (Illustration Credit bm1.21) Quarternion Julia sets (Illustration Credit bm1.22) Illustration Credits Reprinted from The Fractal Geometry of Nature: 29.1 Courtesy Michael Frame: aft.1, aft.3 Augusto Giacometti, 1912: bm1.

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20 bm1.21 Courtesy E. R. Weibel, Institute of Anatomy, University of Bern: bm1.13 ABOUT THE AUTHOR A graduate of the École Polytechnique, Benoit Mandelbrot received his doctorate in mathematics from the University of Paris and spent thirty-five years at IBM as a research scientist and seventeen years as

Mapmatics: How We Navigate the World Through Numbers

by Paulina Rowinska  · 5 Jun 2024  · 361pp  · 100,834 words

t seem to realize how world-shaking his discovery was and didn’t publish his results. So, nobody noticed – almost. Noisy cauliflowers From Benoit B. Mandelbrot’s CV, it’s hard to infer that he would become one of the most important twentieth-century mathematicians. He earned his tenure only after

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Benoit, his uncle Szalom Mandelbrojt, a mathematics professor already living in this unfamiliar country, took care of his education. Despite being mostly self-taught, young Mandelbrot passed the entry exams to the prestigious École Normale in Paris. However, he left the university after one day, unhappy with the faculty’s contempt

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, where he became interested in a special type of pattern that comes up in fields as diverse as abstract mathematics, biology and economics. At IBM, Mandelbrot was tasked with eliminating the unwanted disruptions – called the noise – from computer data transmissions over phone lines. The company thought that these errors were scattered

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at random, but Mandelbrot noticed that they appeared in clusters. When plotted, these clusters revealed an interesting property. A picture representing all the noise within one day was almost

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market. You can keep going, although at some point the tiny branches will resemble flour rather than the initial cauliflower. In his 2010 TED talk, Mandelbrot described a cauliflower as very complicated and very simple at the same time. Smaller pieces of this vegetable are similar to the whole – not identical

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the company: the noise wasn’t generated by guys with screwdrivers messing around with the network, as IBM’s engineers tended to explain the problem. Mandelbrot understood that a ‘guy with a screwdriver’ wouldn’t create such a consistent pattern of disruptions. Instead, the self-similarity indicated a structural problem rather

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by the scientific community was finally recognized for what it was – evidence that nature doesn’t care about the standard, school-taught geometry. For years, Mandelbrot had studied intriguing, self-similar structures such as transmission noise, but fellow scientists didn’t take his research seriously. Whenever he suggested that some process

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newfangled techniques, for which the only references are dusty tomes written in French, or even in Polish, or incomprehensible modern monographs?’ In the coastline paradox, Mandelbrot saw a concrete and visual example of the complicated notion of self-similarity he’d struggled to convince people of for a long time. In

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1967, Mandelbrot followed up on Richardson’s results, this time in a journal as well-known as it gets: Science. Because every scientist worth her or his

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salt regularly glances at Science, Richardson’s revolutionary insights finally saw daylight. While the title of Mandelbrot’s paper wasn’t too imaginative (although we’ve already seen worse), ‘How Long Is the Coast of Britain?’ piqued the interest of fellow scientists

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it. And there was. The subtitle ‘Statistical Self-Similarity and Fractional Dimension’ revealed what this paper was about. Mandelbrot, taking Richardson’s work as an example, introduced the concept of self-similarity and fractality (which is pretty much a fancy name for the wiggliness of coastlines and borders) to the wider world

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than the elephant’s. But the coast of Britain, unlike the coast of a circular island, isn’t one-dimensional. In his seminal Science paper, Mandelbrot suggested a different way of looking at ‘dimensions’ and made an appeal to stop using the notion of length for such non-ordinary curves altogether

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the measuring stick but also on a special number D assigned to each curve. Richardson didn’t think much of this unitless number. Later, however, Mandelbrot figured it could be treated as a special type of dimension that characterizes the curve’s wiggliness, a bit like how density assigned to each

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times if the scale doubles, 3D times if the scale triples, etc. He didn’t, however, provide any physical interpretation of this number. It was Mandelbrot who recognized that D can be interpreted as a dimension and coined the term fractional dimension. While similar concepts had been used to describe theoretical

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it was the first time someone connected the abstract notion of dimension with real-world phenomena, from coastlines to computer noise. ‘A Trojan horse manoeuver’ Mandelbrot’s revolutionary idea didn’t fly with the mathematics community. He was ostracized, and it’s hard to blame the mathematicians for their scepticism. We

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exactly what even the most open-minded mathematicians thought not such a long time ago. In the annotations to his Science paper, Mandelbrot complained that ‘mention of a fractal dimension in a paper or a talk led all referees and editors to their pencils, and some audiences to audible signs of disapproval

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. Practitioners accused me of hiding behind formulas that were purposefully incomprehensible’. Only the accidental discovery of Richardson’s appendix helped Mandelbrot to change the general

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response. If borders and coasts – curves we know and understand – have these weird fractal dimensions, maybe the concept of a fractal dimension isn’t as abstract as it seems? By disguising a deep mathematical

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theory behind a geographical curiosity – a ‘Trojan horse manoeuver’, as he called it himself – Mandelbrot brought fractal dimensions to a wider audience. Given the usual unenthusiastic approach to his work, the prompt acceptance of his paper submission to Science came as a

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surprise. The referee behind the positive feedback was the esteemed Polish mathematician, Hugo Steinhaus. Years before Mandelbrot, Steinhaus had already noticed the coastline paradox when measuring

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as the length read off the school map’. No wonder that when Steinhaus saw his own insights, also completely ignored by the mathematical community, in Mandelbrot’s paper, he promptly recommended that the paper should be published. He must have seen it as a chance for the coastline paradox, known from

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ship would sail along a smooth, almost circular path, while the smaller boat would follow the island’s bays and inlets, increasing the reported length. Mandelbrot wondered why this omnipresent measurement problem had been consistently ignored, especially given that Steinhaus, a world-class mathematician, had studied it in his paper. He

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‘the fact that no one reads Polish scientific journals – even those written in an international language’ (a conclusion which would leave me personally offended). In Mandelbrot’s opinion, Steinhaus himself hadn’t considered his insights worthy of publication; they had found their way onto the printing press only because of the

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the only case where the problematic ‘publish or perish’ approach, so common in academia, has resulted in something positive. Maybe Mandelbrot wasn’t the first to notice the need for fractal dimensions, but he was the first to understand its importance. Being an excellent science communicator, he managed not only to popularize

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had less memory than today’s smartphones. Here’s where Carpenter comes in. He stumbled upon the book The Fractal Geometry of Nature, in which Mandelbrot had described how fractals work and included pictures of fractally generated landscapes. This gave Carpenter a brilliant idea: instead of storing the information about all parts of a

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mountain range, the machine would need only the fractal pattern to visualize the whole landscape. By definition, a fractal is a repetition of

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movie look realistic, you can safely assume that the secret behind them is fractal-based software, RenderMan or similar. From Forrest Gump’s incredible ping-pong game to realistic dinosaurs in Jurassic Park and icy landscapes in Frozen, Mandelbrot’s insights changed cinematography. RenderMan became the first software to receive an Oscar

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– or four Oscars, to be precise. One of the statues went to the three men behind the fractal-based software, Loren Carpenter and his Pixar colleagues Ed Catmull

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for divergences from a healthy, fractal heartbeat or distinguish normal tissue from a non-fractal tumour. Meanwhile, engineers build fractal antennas for the best signal strength to area ratio, mimicking the optimal, fractal shapes designed by nature: from our lungs to trees, to river deltas, to DNA folds. As Mandelbrot famously said, ‘clouds are not

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spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line’. Once you see it, you won’t be able to unsee it. Unfortunately, the failure to notice fractals can have dire

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. Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies. New York: Penguin Press, 2017. Benoit Mandelbrot and Fractals Gleick, James. Chaos: Making a New Science. New York: Viking, 1987. Lesmoir-Gordon, Nigel, ed. The Colours of Infinity: The Beauty and Power of

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Fractals, 2nd ed. London: Springer, 2010. Mandelbrot, Benoît. ‘Drawing; The Ability to Think in Pictures and Its Continuing Influence’. Web of Stories videos, 24 January 2008, https://www.webofstories.com

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of Stories video, 24 January 2008, https://www.webofstories.com/play/benoit.mandelbrot/16. (and getting married in the meantime): Nigel Lesmoir-Gordon, ‘Benoît Mandelbrot Obituary’, The Guardian, 17 October 2010. very complicated and very simple at the same time: Benoit Mandelbrot, ‘Fractals and the Art of Roughness’, TED Talks video, February 2010, https

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://www.ted.com/talks/benoit_mandelbrot_fractals_and_the_art_of_roughness?language=en. no possible

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significance: Benoît Mandelbrot, ‘Errors of Transmission in Telephone Channels (50/144)’, Web of Stories

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Review of Fluid Mechanics 30, no. 1 (January 1998): xiii–xxxvi, https://doi.org/10.1146/annurev.fluid.30.1.0. incomprehensible modern monographs?: ‘Benoit Mandelbrot’, interview by Anthony Barcellos, in Mathematical People: Profiles and Interviews, ed. Donald J. Albers and Gerald L. Alexanderson, 2nd ed. (Wellesley, MA: A. K.

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over Rio de Janeiro from ‘41204668869’: S2 Region Coverer Online Viewer, accessed 4 February 2024, https://igorgatis.github.io/ws2/?cells=00997fd59c5. purposefully incomprehensible: Benoit Mandelbrot, ‘How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension’, Science 156, no. 3775 (5 May 1967): 636–8, https://doi.org

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/10.1126/science.156.3775.636. Mandelbrot brought fractal dimensions to a wider audience: Mandelbrot, ‘How Long Is the Coast of Britain?’. the length read off the school map: Hugo Steinhaus, ‘Length, Shape and Area’, Colloquium Mathematicum

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, https://doi.org/10.4064/cm-3-1-1-13. Sicilian fields were known to have bigger areas: Benoît Mandelbrot, ‘A Geometry Able to Include Mountains and Clouds’, in The Colours of Infinity: The Beauty, the Power of Fractals, ed. Nigel Lesmoir-Gordon (London: Springer, 2010), 38–57. leave me personally offended

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: Mandelbrot, ‘How Long Is the Coast of Britain?’. a mountain behind it: ‘Hunting the Hidden Dimension’, NOVA Transcripts, Public Broadcasting Service

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, 28 October 2008, https://www.pbs.org/wgbh/nova/transcripts/3514_fractals.html. the results stunned even him: Rachel Sullivan, ‘Hiding in

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Renderman?’, Pixar, n.d., accessed 7 September 2021, https://renderman.pixar.com/about. nor does lightning travel in a straight line: Benoit B. Mandelbrot, ‘Introduction: Theme’, in The Fractal Geometry of Nature (San Francisco: W. H. Freeman, 1983; New York: W. H. Freeman, 2021), 1–5. the US and Russian territories: ‘How

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ref1 Fossett, Steve ref1, ref2 Foster, Howard ref1 four-colour theorem ref1, ref2, ref3, ref4, ref5, ref6 Four Corners Monument, USA ref1n Fractal Geometry of Nature, The (Mandelbrot) ref1 fractals ref1, ref2, ref3, ref4 Fréchet, Maurice René ref1 French Academy of Sciences ref1 Frisius, Gemma ref1, ref2, ref3n functional magnetic resonance imaging (fMRI

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History of the Life and Voyages of Christopher Columbus (Irving) ref1 homeomorphisms ref1, ref2 Horserød camp, Denmark ref1n ‘How Long is the Coast of Britain?’ (Mandelbrot) ref1, ref2, ref3 Hungary, gerrymandering in ref1 IBM ref1 Ibn Sahl ref1n InSight lander (NASA) ref1 integrals ref1 interior of the Earth, mapping the ref1

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ref1 loxodromes ref1 Lucasfilm ref1 Ma, Stephen ref1 McGhee, Eric ref1, ref2, ref3 Madison, James ref1 magnetic field, Earth’s ref1, ref2n Mandelbrojt, Szalom ref1 Mandelbrot, Benoit B. ref1, ref2, ref3, ref4, ref5, ref6 Manhattan metric ref1, ref2 mantle plumes ref1 mapmaking areas, preserving ref1, ref2 distances, preserving ref1 Gall–Peters

The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

by Mario Livio  · 23 Sep 2003

that it did embody. Most recently, in 1999, French author and telecommunications expert Midhat J. Gazalé writes in his interesting book Gnomon: From Pharaohs to Fractals: “It was reported that the Greek historian Herodotus learned from the Egyptian priests that the square of the Great Pyramid's height is equal to

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and symmetry in solids. Once I stumbled on the problem of quasi-periodic crystals, I found it irresistible and I kept coming back to it.” FRACTALS The Steinhardt-Jeong model for quasi-crystals has the interesting property that it produces long-range order from neighborly interactions, without resulting in a fully

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property, like the Russian Matrioshka dolls that fit one into the other, are known as, fractals. The name “fractal” (from the Latin fractus, meaning “broken, fragmented”) was coined by the famous Polish-French-American mathematician Benoit B. Mandelbrot, and it is a central concept in the geometry of nature and in the theory

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of highly irregular systems known as chaos. Fractal geometry represents a brilliant attempt to describe the shapes and objects of the real

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), a painting by the German romantic painter Caspar David Friedrich (1774–1840; currently in the Gemäldegalerie Neue Meister in Dresden). Figure 111 Mandelbrot's gigantic mental leap in formulating fractal geometry has been primarily in the fact that he recognized that all of these complex zigs and zags are not merely a

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nuisance but often the main mathematical characteristic of the morphology. Mandelbrot's first realization was the importance of self-similarity—the fact that many

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does not smooth out the degree of roughness. Rather, the same irregularities characterize all scales. At this point, Mandelbrot asked himself, how do you determine the dimensions of something that has such a fractal structure? In the world of Euclidean geometry, all the objects have dimensions that can be expressed as whole

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lines are one-dimensional, plane figures like triangles and pen tagons are two-dimensional, and objects like spheres and the Platonic solids are three-dimensional. Fractal curves like the path of a bolt of lightning, on the other hand, wiggle so aggressively that they fall somewhere between one and two dimensions

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. If the path is relatively smooth, then we can imagine that the fractal dimension would be close to one, but if it is very complex, then a dimension closer to two can be expected. These musings have turned

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into the by now-famous question: “How long is the coast of Britain?” Mandelbrot's surprising answer is that the length of the coastline actually depends on the length of your ruler. Suppose you start out with a satellite

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exists substructure on even smaller scale. This fact suggests that even the concept of length as representing size needs to be revisited when dealing with fractals. The contours of the coastline do not become a straight line upon magnification; rather, the crinkles persist on all scales and the length increases ad

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-fifths that of the original triangle.) Figure 113 The realization of the existence of fractals raised the question of the dimensions that should be associated with them. The fractal dimension is really a measure of the wrinkliness of the fractal, or of how fast length, surface, or volume increases if we measure it

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two-dimensional square. But how can it have an intermediate dimension? There is, after all, no whole number between 1 and 2. This is where Mandelbrot followed a concept first introduced in 1919 by the German mathematician Felix Hausdorff (1868–1942), a concept that at first appears mind boggling—fractional dimensions

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such a notion, fractional dimensions were precisely the tool needed to characterize the degree of irregularity, or fractal complexity, of objects. In order to obtain a meaningful definition of the self-similarity dimension or fractal dimension, it helps to use the familiar whole-number dimensions 0, 1, 2, 3 as guides. The

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this relation in Appendix 7.) Applying the same relation to the Koch snowflake gives a fractal dimension of about 1.2619. As it turns out, the coastline of Britain also has a fractal dimension of about 1.26. Fractals therefore serve as models for real coastlines. Indeed, pioneering chaos theorist Mitch Feigenbaum, of

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this fact to help produce in 1992 the revolutionary Hammond Atlas of the World. Using computers to do as much as possible unassisted, Feigenbaum examined fractal satellite data to determine which points along coastlines have the greatest significance. The result—a map of South America, for example, that is better than

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98 percent accurate, compared to the more conventional 95 percent scored by older atlases. Figure 114 For many fractals in nature, from trees to the growth of crystals, the main characteristic is branching. Let us examine a highly simplified model for this ubiquitous phenomenon

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.618.… (A short proof is given in Appendix 8.) This is known as a Golden Tree, and its fractal dimension turns out to be about 1.4404. The Golden Tree and similar fractals composed of simple lines cannot be resolved very easily with the naked eye after several iterations. The problem can

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resulting image, a Golden Tree composed of lunes, is shown in Figure 118. Figure 117 Figure 118 Figure 119 Figure 120 Figure 121 Figure 122 Fractals can be constructed not just from lines but also from simple planar figures such as triangles and squares. For example, you can start with an

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touch, as in Figure 120, and again the answer turns out to be 1/ φ. Precisely the same situation occurs if you build a similar fractal using a square (Figure 121)—overlapping occurs when the reduction factor is 1/ φ = 0.618… (Figure 122). Furthermore, all the unfilled white rectangles in

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the last Figure are Golden Rectangles. We therefore find that while in Euclidean geometry the Golden Ratio originated from the pentagon, in fractal geometry it is associated even with simpler figures like squares and equilateral triangles. Once you get used to the concept, you realize that the world

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around us is full of fractals. Objects as diverse as the profiles of the tops of forests on the horizon and the circulatory system in a kidney can be described in

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terms of fractal geometry. If a particular model of the universe as a whole known as eternal inflation is correct, then even the entire universe is characterized by

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a fractal pattern. Let me explain this concept very briefly, giving only the broad-brush picture. The inflationary theory, originally advanced by Alan Guth, suggests that when

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, producing a pocket universe to become the same size Figure because of space constraints). An infinite number of pocket universes thus were produced, and a fractal pattern was generated—the same sequence of false vacua and pocket universes is replicated on ever-decreasing scales. If this model truly represents the evolution

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number of pocket universes that exist. Figure 123 Figure 124 In 1990, North Carolina State University professor Jasper Memory published a poem entitled “Blake and Fractals” in the Mathematics Magazine. Referring to the mystic poet William Blake's line “To see a World in a Grain of Sand,” Memory wrote: William

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infinity In the smallest speck of sand Held in the hollow of his hand. Models for this claim we've got In the work of Mandelbrot: Fractal diagrams partake Of the essence sensed by Blake. Basic forms will still prevail Independent of the Scale; Viewed from far or viewed from near Special

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there, Rich in structure all the way— Just as the mystic poets say. Some of the modern applications of the Golden Ratio, Fibonacci numbers, and fractals reach into areas that are much more down to earth than the inflationary model of the universe. In fact, some say that the applications can

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curve is a reduced-scale version of the whole, a concept central to fractal geometry. Indeed, in 1997, Benoit Mandelbrot published a book entitled Fractals and Scaling in Finance: Discontinuity, Concentration, Risk, which introduced well-defined fractal models into market economics. Mandelbrot built on the known fact that fluctuations in the stock market look the

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allow you to read the scales, you will not be able to tell if it represents daily, weekly, or hourly variations. The main innovation in Mandelbrot's theory, as compared to standard portfolio theory, is in its ability to reproduce tumultuous trading as well as placid markets. Portfolio theory, on the

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other hand, is able to characterize only relatively tranquil activity. Mandelbrot never claimed that his theory could predict a price drop or rise on a specific day but rather that the model could be used to

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estimate probabilities of potential outcomes. After Mandelbrot published a simplified description of his model in Scientific American in February 1999, a myriad of responses from readers ensued. Robert Ihnot of Chicago probably

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we know that a stock will go from $10 to $15 in a given amount of time, it doesn't matter how we interpose the fractals, or whether the graph looks authentic or not. The important thing is that we could buy at $10 and sell at $15. Everyone should now

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was in the sequence, the closer it came to the corresponding power of 1.13198824 …. To actually calculate this strange number, Viswanath had to use fractals and to rely on a powerful mathematical theorem that was formulated in the early 1960s by mathematicians Hillel Furstenberg of the Hebrew University in Jerusalem

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. Somehow the Golden Ratio always makes an unexpected appearance at the juxtaposition of the simple and the complex, at the intersection of Euclidean geometry and fractal geometry. The sense of gratification provided by the Golden Ratio's surprising emergences probably comes as close as we could expect to the sensuous visual

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the Platonic ideas. This “modified Platonic view” argues that the laws of physics are expressed as mathematical equations, the structure of the universe is a fractal, galaxies arrange themselves in logarithmic spirals, and so on, because mathematics is the universe's language. Specifically, mathematical objects are still assumed to exist objectively

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. Gleick, J. Chaos. New York: Penguin Books, 1987. Lesmoir-Gordon, N., Rood, W, andEdney, R. Introducing Fractal Geometry. Cambridge: Icon Books, 2000. Mandelbrot, B.B. Fractal Geometry of Nature. New York: W H. Freeman and Company, 1988. Mandelbrot, B.B. “A Multifractal Walk Down Wall Street,” Scientific American (February 1999): 70–73. Matthews, R

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. “The Power of One,” New Scientist, July 10, 1999, 27–30. Peitgen, H.-O., Jürgens, H., andSaupe, D. Chaos and Fractals. New York: Springer-Verlag, 1992. Peterson, I

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,” Science News, 155 (1999): 60–61 Prechter, R.R. Jr., and Frost, A.J. Elliot Wave Principle. Gainesville, GA: New Classics Library, 1978. Schroeder, M. Fractals, Chaos, Power Laws. New York: W H. Freeman and Company, 1991. Steinhardt, P.J., Jeong, H.-C, Saitoh, K., Tanaka, M., Abe, E., andTsai, A

More Than You Know: Finding Financial Wisdom in Unconventional Places (Updated and Expanded)

by Michael J. Mauboussin  · 1 Jan 2006  · 348pp  · 83,490 words

in a pattern that conforms to a standard bell curve. Do financial data neatly conform to such assumptions? Of course, they never do. —Benoit B. Mandelbrot, “A Multifractal Walk down Wall Street” The very fact that the Petersburg Problem has not yielded a unique and generally acceptable solution to more than

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, and the small branches resemble the big branches. These systems are fractal. Unlike a normal distribution, no average value adequately characterizes a fractal system. Exhibit 32.1 contrasts normal and fractal systems visually and shows the probability functions that represent the data. Fractal systems follow a power law.4 EXHIBIT 32.1 Probability Density

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Functions for Normal and Fractal Systems Source: Liebovitch and Scheurle, “Two Lessons from

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Fractals and Chaos.” Reproduced with permission. Using the statistics of normal

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distributions to characterize a fractal system like financial markets is potentially very hazardous. Yet theoreticians and practitioners do it

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daily.5 The distinction between the two systems boils down to probabilities and payoffs. Fractal systems have few, very large observations that fall outside the normal distribution. The classic example is the crash of 1987. The probability (assuming a normal

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billion payoff, but this is only a 1-in-1.1 billion probability. Lots of small events and a few very large events characterize a fractal system. Further, the average winnings per game is unstable with the St. Petersburg game, so no average accurately describes the game’s long-term outcome

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. Are stock market returns fractal? Benoit Mandelbrot shows that by lengthening or shortening the horizontal axis of a price series—effectively speeding up or slowing down time—price series are indeed

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fractal. Not only are rare large changes interspersed with lots of smaller ones, the price changes look similar at various scales (e.g., daily, weekly, and

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monthly returns). Mandelbrot calls financial time series multifractal, adding the prefix “multi” to capture

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the time adjustment. EXHIBIT 32.3 Fractal Coin Toss Game Source: Author analysis. In an important and fascinating book, Why Stock Markets Crash, geophysicist

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’s law does describe some systems well, but is too narrow to describe the variety of systems that exhibit power laws. The brilliant polymath Benoit Mandelbrot showed that two modifications to Zipf’s law make it possible to obtain a more general power law.5 The first modification is to add

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of Pollock’s paintings, physicist Richard Taylor turned to the world of mathematics. He found that Pollock’s paintings, while seemingly haphazard, exhibit pleasing fractal patterns. A fractal is “a geometric shape that can be separated into parts, each of which is a reduced-scale version of the whole.”3 In spite

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of the skeptical sneers, Taylor showed that fractal patterns are by no means an inevitable consequence of dripping paint. Fractals are ubiquitous in nature—trees, clouds, and coastlines are but a few examples—and as a result are visually

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familiar to humans.4 One critical feature of a fractal pattern is its fractal dimension—or degree of complexity (a line has a

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fractal dimension of 1.0, while a filled space has a dimension of 2.0). Taylor and his collaborators found

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that humans have a preference for fractals with dimensions between 1.3 and 1.5, whether those fractals are natural or human-made. Many of Pollock’s paintings fall within, or near, this range. As a consequence, scientists

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8, 1949. Source: Collection Neuberger Museum of Art, Purchase College, State University of New York, gift of Roy R. Neuberger. Photo by Jim Frank. Because fractals are so common in nature, scientists often associate them with self-organized systems. Since economics deals largely with these types of systems, we might expect

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to see fractals in economic systems as well. And indeed, we do. Just as we have to analyze a Pollock painting or a coastline to appreciate the underlying

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fractal pattern, we must take a fresh look at economic systems as well. Order is often hidden.6 Stairway to Shareholder Heaven Self-affinity, or the

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resemblance of the parts to the whole, is another crucial feature of a fractal. Think of a cauliflower. The whole cauliflower, a large bump, and a small bump all visually resemble one another. Stock price changes are also

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fractal: after some adjustments, the data look the same whether you look at month-to-month, week-to-week, or day-to-day changes.7 Analysis

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tradeoff between risk and reward. Unfortunately, the empirical record defies a simple risk-reward relationship. As Benoit Mandelbrot has argued, failure to explain is caused by failure to describe.Starting in earnest with Mandelbrot’s work in finance in the early 1960s, statistical studies have shown that stock price changes are not

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bombardment of heat-excited water molecules on the pollen. 2 See GloriaMundi, “Introduction to VaR,” http://www.gloriamundi.org/introduction.asp. 3 Edgar E. Peters, Fractal Market Analysis (New York: John Wiley & Sons, 1994), 21-27. 4 Roger Lowenstein, When Genius Failed: The Rise and Fall of Long-Term Capital Management

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. See http://www.ccs.fau.edu/˜liebovitch/complexity-20.html. 4 See chapter 29. 5 Benoit B. Mandelbrot, “A Multifractal Walk down Wall Street,” Scientific American, February 1999, 70-73. Also see, Benoit B. Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk (New York: Springer Verlag, 1997). 6 If you assume that

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. 2 One example is the children’s book character, Olivia. See Ian Falconer, Olivia (New York: Atheneum Books for Young Readers, 2000). 3 Benoit B. Mandelbrot, “A Multifractal Walk Down Wall Street,” Scientific American (February 1999): 71. 4 Richard P. Taylor, B. Spehar, C.W.G. Clifford, and B.R. Newell

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, “The Visual Complexity of Pollock’s Dripped Fractals,” Proceedings of the International Conference of Complex Systems, 2002, http://materialscience.uoregon.edu/taylor/art/TaylorlCCS2002.pdf. 5 Richard P. Taylor, “Order in Pollock’s

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the Growth Dynamics of Complex Organizations,” Physical Review Letters 81, no. 15 (October 1998): 3275-3278, http://polymer.bu.edu/hes/articles/lacms98.pdf. 7 Mandelbrot, “A Multifractal Walk Down Wall Street.” Stock price changes are more accurately described as multifractal. Multifractals accommodate some adjustments to get to statistical similarity on

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Biases, ed. Daniel Kahneman, Paul Slovic, and Amos Tversky, 306-34. Cambridge: Cambridge University Press, 1982. Liebovitch, Larry S., and Daniela Scheurle. “Two Lessons from Fractals and Chaos.” Complexity 5, no. 4 (2000): 34-43. Lipshitz, Raanan, Gary Klein, Judith Orasanu, and Eduardo Salas. “Taking Stock of Naturalistic Decision Making.” Working

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York: W. W. Norton, 2003. ——. “Returns from Investing in Equity Mutual Funds, 1971-1991.” Journal of Finance 50, no. 2 (June 1995): 549-72. Mandelbrot, Benoit B. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. New York: Springer Verlag, 1997. ——. “A Multifractal Walk Down Wall Street.” Scientific American (February 1999): 70-73

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E., The Difference: How the Power of Diversity Creates Better Groups, Firms, Schools, and Societies. Princeton, N.J.: Princeton University Press, 2007. Peters, Edgar E. Fractal Market Analysis. New York: Wiley, 1994. Pinker, Steven. The Language Instinct: How the Mind Creates Language. New York: HarperCollins, 1994. Poundstone, William. Prisoner’s Dilemma

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.edu/taylor/art/scientificamerican.pdf. Taylor, Richard P., B. Spehar, C.W.G. Clifford, and B.R. Newell, “The Visual Complexity of Pollock’s Dripped Fractals,” Proceedings of the International Conference of Complex Systems, 2002, http://materialscience.uoregon.edu/taylor/art/TaylorlCCS2002.pdf. Tetlock, Philip E. Expert Political Judgment: How Good

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Simulation of Social Agents: Architectures and Institutions, Argonne National Laboratory and University of Chicago, October 2000, Argonne 2001, 33-51. Mandelbrot, Benoit, and Richard L. Hudson. The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books, 2004. Rothschild, Michael. Bionomics. New York: Henry Holt and

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Company, 1990. Schroeder, Manfred. Fractals, Chaos, and Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, 1991. Seeley, Thomas A., P. Kirk Visscher, and Kevin M. Passino. “

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; short vs. long jumps; types of flight simulators focus; short-term.See also long term, management for Fooled by Randomness (Taleb) Fortune 50, Foster, Richard fractal systems French, Kenneth frequencies; magnitude vs. fruit flies (Drosophila melanogaster) fundamental analysis Galton, Francis gambling Gates, Bill Gazzaniga, Michael General Electric General Theory of Employment

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equity-risk premium exhibits myopic portfolio turnover ratio of risk to reward utility lottery players Lowenstein, Roger luck MacGregor, Donald G. MacKay, Charles Malkiel, Burton Mandelbrot, Benoit B. market capitalization markets: bubbles and crashes collective decisions and decision effect of psychology on efficiency of innovation considered by interpreting new entrants and

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evaluation large cap leveraged performance vs. percentage of outperforming stocks portfolio turnover costs loss aversion stress and positive feedback power laws company-size distribution and fractals and investor understanding of Zipf’s law predictability, loss of prediction price changes, press reports and price-earnings ratios (P/Es) bounded parameters growth and

NumPy Cookbook

by Ivan Idris  · 30 Sep 2012  · 197pp  · 35,256 words

, Winding Along with IPython Combining images In this recipe, we will combine the famous Mandelbrot fractal (for more information on Madelbrot set visit http://en.wikipedia.org/wiki/Mandelbrot_set) and the image of Lena. These types of fractals are defined by a recursive formula, where you calculate the next complex number in a

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a reference to the related recipe. How to do it... We will start by initializing the arrays, followed by generating and plotting the fractal, and finally, combining the fractal with the Lena image. Initialize the arrays.We will initialize x, y, and z arrays corresponding to the pixels in the image area

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.linspace(y_min, y_max, SIZE)) c = x + 1j * y z = c.copy() fractal = numpy.zeros(z.shape, dtype=numpy.uint8) + MAX_COLOR Generate the fractal.If z is a complex number, you have the following relation for a Mandelbrot fractal: In this equation, c is a constant complex number. This can be graphed

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in the complex plane with horizontal real values axis and vertical imaginary values axis. We will use the so-called "escape time algorithm" to draw the fractal. The algorithm scans the points

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.abs(z) <= 4 z[mask] = z[mask] ** 2 + c[mask] fractal[(fractal == MAX_COLOR) & (-mask)] = (MAX_COLOR - 1) * n / ITERATIONS Plot the fractal.Plot the fractal with Matplotlib: matplotlib.pyplot.subplot(211) matplotlib.pyplot.imshow(fractal) matplotlib.pyplot.title('Mandelbrot') matplotlib.pyplot.axis('off') Combine the fractal and Lena.Use the choose function to pick value from

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the fractal or Lena array: matplotlib.pyplot.subplot(212) matplotlib.pyplot.imshow

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(numpy.choose(fractal < lena, [fractal, lena])) matplotlib.pyplot.axis('off') matplotlib.pyplot.title('Mandelbrot + Lena') The following is

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the resulting image: The following is the complete code for this recipe: import numpy import matplotlib.pyplot import sys import scipy if(len(sys.argv) != 2): print "Please input the number of iterations for the fractal" sys.exit() ITERATIONS = int(sys

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(ITERATIONS): print n mask = numpy.abs(z) <= 4 z[mask] = z[mask] ** 2 + c[mask] fractal[(fractal == MAX_COLOR) & (-mask)] = (MAX_COLOR - 1) * n / ITERATIONS # Display the fractal matplotlib.pyplot.subplot(211) matplotlib.pyplot.imshow(fractal) matplotlib.pyplot.title('Mandelbrot') matplotlib.pyplot.axis('off') # Combine with lena matplotlib.pyplot.subplot(212) matplotlib.pyplot.imshow(numpy

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.choose(fractal < lena, [fractal, lena])) matplotlib.pyplot.axis('off') matplotlib.pyplot.title('Mandelbrot + Lena') matplotlib.pyplot.show() How it works... The following functions were used in this example: Function Description linspace Returns numbers within a range with a

Statistical Arbitrage: Algorithmic Trading Insights and Techniques

by Andrew Pole  · 14 Sep 2007  · 257pp  · 13,443 words

Structural Models Introduction Formal Forecast Functions Exponentially Weighted Moving Average Classical Time Series Models Autoregression and Cointegration Dynamic Linear Model Volatility Modeling Pattern Finding Techniques Fractal Analysis Which Return? A Factor Model Factor Analysis Defactored Returns Prediction Model Stochastic Resonance Practical Matters Doubling: A Deeper Perspective Factor Analysis Primer Prediction Model

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early 1990s. Statistical arbitrage approaches range from the vanilla pairs trading scheme of old to sophisticated, dynamic, nonlinear models employing techniques including neural networks, wavelets, fractals— just about any pattern matching technology from statistics, physics, and mathematics has been tried, tested, and in a lot of cases, abandoned. Later developments combined

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help uncover persistent changes in the nature of the reversion phenomenon exploited by a model. Reversion is exhibited by stock price spreads on many frequencies (Mandelbrot, fractal analysis), one of which is targeted by a modeler’s chosen calibration, that choice being dictated by factors including robustness of the response to small

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existence fleeting in terms of usable exploitation opportunities. Borrowing from Orwell’s fancy: description yes, prediction no. 3.4.5 Fractal Analysis We refer the interested reader to the inventor, Benoit B. Mandelbrot 2004, who tells it best. 3.5 WHICH RETURN? Which return do you want to forecast? The answer may

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the variation of the spread overall. Such self-similarity occurs throughout nature according to Benoit Mandelbrot, who invented a branch of mathematics called fractals for the study and analysis of such patterns. Mandelbrot, 2004, has argued that fractal analysis provides a better model for understanding the movements of prices of financial instruments than anything

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currently in the mathematical finance literature. It is unknown whether any successful trading strategies have been built using fractal analysis; Mandelbrot himself does not believe his tools are yet sufficiently developed for prediction of financial series to be feasible. 3.8 PRACTICAL MATTERS Forecasts of

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to zero, solve). 4.2.5 Generalizations Financial time series are notorious for the tenacity with which they refuse to reveal underlying mathematical structure (though Mandelbrot, 2004, may demur from that statement). Features of such data, which often show up in statistical modeling, include: nonnormal distributions (returns are frequently characterized by

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leptokurtosis); nonconstant variance (market dynamics often produce bursts of high and low volatility, and modelers have tried many approaches from GARCH and its variants to Mandelbrot’s fractals, see Chapter 3); and serial dependence. The conditions of the theorem can be relaxed to accommodate all of these behaviors. 74 STATISTICAL ARBITRAGE The

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Wiley & Sons, 1994. Lehman Brothers. Algorithmic Trading. New York: Lehman Brothers, 2004. Mandelbrot, B.B. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. New York: Springer-Verlag, 1997. Mandelbrot B.B., and R.L. Hudson. The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books, 2004. Orwell

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, 126 Change point identification, 200 Chi-square distribution, 96 Classical time series models, 47–52 autoregression and cointegration, 47–49 dynamic linear model, 49–50 fractal analysis, 52 pattern finding techniques, 51–52 volatility modeling, 50–51 225 226 Cointegration and autoregression, 47–49 Competition, return decline and, 160–162 Conditional

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) spread, 101–102, 106–107 Forecast monitor, 42–43, 172 Forecasts: with catastrophe process, 198–200 signal noise and, 167–174 Forecast variance, 26–28 Fractals, 52, 59, 73 Inverse Gamma distribution, 76–77 Iraq, U.S. invasion of, 175–176, 179 Gamma distribution, 96 GARCH (generalized autoregressive conditional heteroscedastic), 50

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–71 Kendall, Maurice, 63 Keynes, J. M., 91 Kidder Peabody, 150 Kotz, S., 134 Managers: performance and correlation, 151–154 relative inactivity of, 166–174 Mandelbrot, Benoit B., 52, 59 Marginal distribution, 86, 93 Market deflation, 189–190 Market exposure, 29–30 Market impact, 30–31, 185–188 Market neutrality, 29

How Big Things Get Done: The Surprising Factors Behind Every Successful Project, From Home Renovations to Space Exploration

by Bent Flyvbjerg and Dan Gardner  · 16 Feb 2023  · 353pp  · 97,029 words

scalewise, which is exactly what you want in order to build something huge with ease. The mathematician Benoit Mandelbrot, who first laid out the science of scale-free scalability, called this attribute “fractal”—like one of those popular Internet memes in which you see a pattern, then zoom into a detail within

…

, so undoubtedly I forgot some, for which I ask forgiveness, but which does not lessen their contribution or my gratitude. Gerd Gigerenzer, Daniel Kahneman, Benoit Mandelbrot, and Nassim Nicholas Taleb are principal intellectual influences. Nobody understands risk better than they do, and understanding risk is the key to understanding big projects

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higher above 3 the kurtosis is for a distribution (called “excess kurtosis”), the more fat-tailed the distribution is considered to be. The mathematician Benoit Mandelbrot found a kurtosis of 43.36 in a pioneering study of daily variations in the Standard & Poor’s 500 index between 1970 and 2001—14

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-tailed than the Gaussian—which he found alarmingly high in terms of financial risk; see Benoit B. Mandelbrot and Richard L. Hudson, The (Mis)behavior of Markets (London: Profile Books, 2008), 96. But Mandelbrot’s finding is not particularly high when compared with the kurtosis I have found for percentage cost overruns

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, Scaling: Why Is Animal Size So Important? (Cambridge, UK: Cambridge University Press, 1984). 10. Author interview with Mike Green, June 5, 2020. 11. Benoit B. Mandelbrot, Fractals and Scaling in Finance (New York: Springer, 1997). 12. Erin Tallman, “Behind the Scenes at China’s Prefab Hospitals Against Coronavirus,” E-Magazine by Medical

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to Floods.” Journal of Hydrology 322 (1–4): 168–80. Manchester Evening News. 2007. “Timeline: The Woes of Wembley Stadium.” Manchester Evening News, February 15. Mandelbrot, Benoit B. 1960. “The Pareto-Lévy Law and the Distribution of Income.” International Economic Review 1 (2): 79–106

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. Mandelbrot, Benoit B. 1963. “New Methods in Statistical Economics.” Journal of Political Economy 71 (5): 421–40. Mandelbrot, Benoit B. 1963. “The Variation of Certain Speculative Prices.” The Journal of Business 36 (4): 394–419

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; correction printed in Mandelbrot, Benoit B. 1972. The Journal of Business 45 (4): 542–43; revised version reprinted in Mandelbrot, Benoit B. 1997. Fractals and Scaling in Finance. New York

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: Springer, 371–418. Mandelbrot, Benoit B. 1997. Fractals and Scaling in Finance. New York: Springer. Mandelbrot, Benoit B., and Richard L. Hudson. 2008. The (Mis)behavior of Markets. London: Profile

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Books. Mandelbrot, Benoit B., and James R. Wallis. 1968. “Noah, Joseph, and Operational Hydrology.” Water Resources Research 4 (5

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Games in, ref1 Los Angeles, California, ref1, ref2, ref3, ref4 Lötschberg Base Tunnel, Switzerland, ref1 Machiavelli factor, ref1 Madrid Metro, ref1, ref2, ref3 Maersk, ref1 Mandelbrot, Benoit, ref1, ref2 Marriott hotel, New York City, ref1 Mass Transit Railway (MTR), Hong Kong, ref1, ref2, ref3, ref4, ref5 masterbuilders, ref1 maximum virtual product

When Computers Can Think: The Artificial Intelligence Singularity

by Anthony Berglas, William Black, Samantha Thalind, Max Scratchmann and Michelle Estes  · 28 Feb 2015

Meat Based Intelligence 1. Silicon vs. neurons 2. Speech understanding 3. Other hardware estimates 4. Small size of genome 5. Chimpanzee intelligence 6. Packing density, fractals, and evolution 7. Repeated patterns 8. Small DNA, small program 7. Related Work 1. Many very recent new books 2. Kurzweil 2000, 2006, 2013 3

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activity. So it may turn out that just a few very special differences in our genotype have resulted in our relatively high intelligence. Packing density, fractals, and evolution The information in genes is tightly packed, with many complex transcription processes. These include using different parts of the same gene to produce

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which allow the neurons to organize themselves. There are mathematical systems that can produce complex artefacts from simple definitions. One well-known example is the Mandelbrot fractal set shown below. One can zoom into this diagram indefinitely and similar, complex, but nonrepeating patterns will be seen

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. Mandelbrot Set Public Wikipedia Amazingly, all this stunning complexity is produced by the following simple equation appropriately interpreted:z’ = z2 + c So if something vaguely analogous to this type of fractal formula could be stored in our DNA, a small amount of

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DNA could result in very complex structures. However, while the Mandelbrot formula can produce this stunningly complex pattern, it cannot produce arbitrary patterns. Moreover, minor changes

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has to be a relatively direct and robust relationship between our genotype and our phenotype. Evolution just could not work with a too highly-packed, fractal-like representation because making any small change to the gene sequence would produce a radically different brain. It would require chancing upon just the right

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The Road to Ruin: The Global Elites' Secret Plan for the Next Financial Crisis

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The Age of Spiritual Machines: When Computers Exceed Human Intelligence

by Ray Kurzweil  · 31 Dec 1998  · 696pp  · 143,736 words

Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets

by Nassim Nicholas Taleb  · 1 Jan 2001  · 111pp  · 1 words

Adaptive Markets: Financial Evolution at the Speed of Thought

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Descartes' Error: Emotion, Reason and the Human Brain

by António R. Damásio  · 1 Jan 1994  · 347pp  · 101,586 words

The Volatility Smile

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The Master and His Emissary: The Divided Brain and the Making of the Western World

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Tools for Computational Finance

by Rüdiger Seydel  · 2 Jan 2002  · 313pp  · 34,042 words

Frequently Asked Questions in Quantitative Finance

by Paul Wilmott  · 3 Jan 2007  · 345pp  · 86,394 words

Coming of Age in the Milky Way

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New Dark Age: Technology and the End of the Future

by James Bridle  · 18 Jun 2018  · 301pp  · 85,263 words

The Sum of All Fears

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The Loop: How Technology Is Creating a World Without Choices and How to Fight Back

by Jacob Ward  · 25 Jan 2022  · 292pp  · 94,660 words

The Vanishing Face of Gaia: A Final Warning

by James E. Lovelock  · 1 Jan 2009  · 239pp  · 68,598 words

Traders, Guns & Money: Knowns and Unknowns in the Dazzling World of Derivatives

by Satyajit Das  · 15 Nov 2006  · 349pp  · 134,041 words

Dawn of the New Everything: Encounters With Reality and Virtual Reality

by Jaron Lanier  · 21 Nov 2017  · 480pp  · 123,979 words

Aurora

by Kim Stanley Robinson  · 6 Jul 2015  · 488pp  · 148,340 words

Overdiagnosed: Making People Sick in the Pursuit of Health

by H. Gilbert Welch, Lisa M. Schwartz and Steven Woloshin  · 18 Jan 2011  · 302pp  · 92,546 words

Only Americans Burn in Hell

by Jarett Kobek  · 10 Apr 2019  · 338pp  · 74,302 words

Is the Internet Changing the Way You Think?: The Net's Impact on Our Minds and Future

by John Brockman  · 18 Jan 2011  · 379pp  · 109,612 words

Planet of Slums

by Mike Davis  · 1 Mar 2006  · 232pp

The Education of a Value Investor: My Transformative Quest for Wealth, Wisdom, and Enlightenment

by Guy Spier  · 8 Sep 2014  · 240pp  · 73,209 words

Shantaram: A Novel

by Gregory David Roberts  · 12 Oct 2004  · 1,222pp  · 385,226 words

Accessory to War: The Unspoken Alliance Between Astrophysics and the Military

by Neil Degrasse Tyson and Avis Lang  · 10 Sep 2018  · 745pp  · 207,187 words

The Great Mental Models: General Thinking Concepts

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Making Sense of Chaos: A Better Economics for a Better World

by J. Doyne Farmer  · 24 Apr 2024  · 406pp  · 114,438 words

Escape From Rome: The Failure of Empire and the Road to Prosperity

by Walter Scheidel  · 14 Oct 2019  · 1,014pp  · 237,531 words

Co-Intelligence: Living and Working With AI

by Ethan Mollick  · 2 Apr 2024  · 189pp  · 58,076 words