Benoit Mandelbrot

Chaos: Making a New Science
by James Gleick
Published 18 Oct 2011

* With a parameter of 3.5, say, and a starting value of .4, he would see a string of numbers like this: .4000, .8400, .4704, .8719, .3908, .8332, .4862, .8743, .3846, .8284, .4976, .8750, .3829, .8270, .4976, .8750, .3829, .8270, .5008, .8750, .3828, .8269, .5009, .8750, .3828, .8269, .5009, .8750, etc. A Geometry of Nature And yet relation appears, A small relation expanding like the shade Of a cloud on sand, a shape on the side of a hill. —WALLACE STEVENS “Connoisseur of Chaos” A PICTURE OF REALITY built up over the years in Benoit Mandelbrot’s mind. In 1960, it was a ghost of an idea, a faint, unfocused image. But Mandelbrot recognized it when he saw it, and there it was on the blackboard in Hendrik Houthakker’s office. Mandelbrot was a mathematical jack-of–all-trades who had been adopted and sheltered by the pure research wing of the International Business Machines Corporation.

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To the physicists expanding on the work of people like Lorenz, Smale, Yorke, and May, this prickly mathematician remained a sideshow—but his techniques and his language became an inseparable part of their new science. The description would not have seemed apt to anyone who knew him in his later years, with his high imposing brow and his list of titles and honors, but Benoit Mandelbrot is best understood as a refugee. He was born in Warsaw in 1924 to a Lithuanian Jewish family, his father a clothing wholesaler, his mother a dentist. Alert to geopolitical reality, the family moved to Paris in 1936, drawn in part by the presence of Mandelbrot’s uncle, Szolem Mandelbrojt, a mathematician.

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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 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 a stubborn question about earthquakes. It had been well known for twenty years that the distribution of large and small earthquakes followed a particular mathematical pattern, precisely the same scaling pattern that seemed to govern the distribution of personal incomes in a free-market economy.

Mapmatics: How We Navigate the World Through Numbers
by Paulina Rowinska
Published 5 Jun 2024

O’Connor and Edmund F. Robertson, ‘Benoit Mandelbrot’, MacTutor (Maths History), School of Mathematics and Statistics, University of St Andrews, July 1999, https://mathshistory.st-andrews.ac.uk/Biographies/Mandelbrot/. the École Polytechnique: Benoît Mandelbrot, ‘École Normale and Thought in Mathematics’, Web 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://www.ted.com/talks/benoit_mandelbrot_fractals_and_the_art_of_roughness?

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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 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/play/benoit.mandelbrot/8. Coastline Paradox Stoa, Ryan B. ‘The Coastline Paradox’. Rutgers University Law Review 72, no. 2 (Winter 2019): 351–400. 4.

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very complicated and very simple at the same time: Benoit Mandelbrot, ‘Fractals and the Art of Roughness’, TED Talks video, February 2010, https://www.ted.com/talks/benoit_mandelbrot_fractals_and_the_art_of_roughness?language=en. no possible significance: Benoît Mandelbrot, ‘Errors of Transmission in Telephone Channels (50/144)’, Web of Stories video, n.d., https://www.youtube.com/watch?v=0EeAclc1OEc. as IBM’s engineers tended to explain the problem: James Gleick, ‘A Geometry of Nature’, in Chaos: Making a New Science (Harrisonburg, VA: Viking, 1987), 81–118. the General Systems Year Book of 1961: Oliver M. Ashford, Prophet or Professor: The Life and Work of Louis Fry Richardson (Bristol: Adam Hilger, 1985), 260, quoted in J.

The Myth of the Rational Market: A History of Risk, Reward, and Delusion on Wall Street
by Justin Fox
Published 29 May 2009

He recommended to his professors that they bring Osborne to the Business School for a semester, but Osborne demurred because his large family made relocation problematic.10 Another early member of the gang was Houthakker. While serving on the Stanford faculty with Holbrook Working in the 1950s, he began to focus on the commodity price series that he had criticized Maurice Kendall for bothering to study. He brought this avocation with him to Harvard, where one day in 1960 Benoit Mandelbrot came calling. Mandelbrot was a mathematician who had emigrated from France to work at IBM’s research center in Yorktown Heights, New York, studying—like Osborne at the Naval Research Laboratory—most anything that interested him. He had been looking at the mathematics of income distribution, and Houthakker invited him up to Harvard to speak about it.

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At Tufts, Fama had crunched numbers for a stock market newsletter published by one of his professors. He found lots of interesting patterns in stock prices, but noticed that they tended to disappear as soon as he had identified them. With this experience he gravitated toward the random walk work begun by statistics professor Harry Roberts. He also hooked up with wandering IBM mathematician Benoit Mandelbrot. His first published work was a Mandelbrot-guided exploration of the statistical distribution of stock price changes. Fama stayed on for his doctorate, and under the influence of the newly arrived Miller he began to steer a course away from purely statistical work toward a research program shaped by economic theory.

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In statistical terms these rare but significant events are called fat tails, because they are found at the tail ends of a statistical distribution and keep them from converging quickly with zero—as they would in a true bell curve. The tendency of fat-tail events to follow upon one another is called dependence. IBM MATHEMATICIAN BENOIT MANDELBROT SAW fat tails and dependence in a chart of cotton futures prices at Harvard in 1960. Mandelbrot was a Polish Jew who had emigrated to France in 1936, spent what would have been his high school years hiding from the Nazis, and then got a doctorate in mathematics at the Sorbonne. It was a 1949 book by Harvard linguist George Zipf that first piqued his interest in strange statistical distributions.

The Quants
by Scott Patterson
Published 2 Feb 2010

This haunting fear, brought on by Black Monday, would hover over them like a bad dream time and time again, from the meltdown in October 1987 until the financial catastrophe that erupted in August 2007. The flaw had already been identified decades earlier by one of the most brilliant mathematicians in the world: Benoit Mandelbrot. When German tanks rumbled into France in 1940, Benoit Mandelbrot was sixteen years old. His family, Lithuanian Jews, had lived in Warsaw before moving to Paris in 1936 amid a spreading economic depression. Mandelbrot’s uncle, Szolem Mandelbrojt, had moved to Paris in 1929 and quickly rose to prominence among the city’s mathematical elite.

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Aaron Brown, the quant who used his math smarts to thoroughly humiliate Wall Street’s old guard at their trademark game of Liar’s Poker, and whose career provided him with a front-row view of the explosion of the mortgage-backed securities industry. Paul Wilmott, quant guru extraordinaire and founder of the mathematical finance program at Oxford University. In 2000, Wilmott began warning of a mathematician-led market meltdown. Benoit Mandelbrot, mathematician who as early as the 1960s warned of the dangers wild market swings pose to quant models—but was soon forgotten in the world of quants as little more than a footnote in their long march to a seemingly inevitable victory. “We have involved ourselves in a colossal muddle, having blundered in the control of a delicate machine, the working of which we do not understand.

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The fine-tuned models, the bell curves and random walks, the calibrated correlations—all the math and science that had propelled the quants to the pinnacle of Wall Street—couldn’t capture what was happening. It was utter chaos driven by pure human fear, the kind that can’t be captured in a computer model or complex algorithm. The wild, fat-tailed moves discovered by Benoit Mandelbrot in the 1950s seemed to be happening on an hourly basis. Nothing like it had ever been seen before. This wasn’t supposed to happen! The quants did their best to contain the damage, but they were like firefighters trying to douse a raging inferno with gasoline—the more they tried to fight the flames by selling, the worse the selling became.

The Fractalist
by Benoit Mandelbrot
Published 30 Oct 2012

Copyright © 2012 by The Estate of Benoit Mandelbrot Afterword copyright © by Michael Frame All rights reserved. Published in the United States by Pantheon Books, a division of Random House, Inc., New York, and in Canada by Random House of Canada Limited, Toronto. Pantheon Books and colophon are registered 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.

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I. Title. QA29.M34A3 2012 510.92—dc22 [B] 2012017896 www.pantheonbooks.com Cover image Benoit Mandelbrot. Emilio Segre Visual Archives/American Institute of Physics/Photo Researchers, Inc. Cover design by Peter Mendelsund v3.1 My long, meandering ride through life has been lonely and often very rough. Without loving help, it would have been short, nasty, and unproductive. But I have been lucky. Father and Mother taught me the art of survival. Uncle took me as an unruly but grateful student. Aliette later joined them, and she, our sons, and our grandchildren taught me how to smile.

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He smiled and answered, “What a surprise. Nice to see you. Of course I remember you.” I breathed again—he would not tell on me. My papers cautiously downgraded my baccalaureate from its dangerously conspicuous summa to an adequate magna. One day, a student approached. “I hear that you come from Tulle. You must have known Benoit Mandelbrot.” “Of course, of course, I know him well.” “Is it true that he is un crack who got a summa at the bachot?” Back in 1944, “crack” was French slang for a high achiever. Imagine my panic. Did the student suspect the truth? Was he testing me? Trembling and with feigned nonchalance, I started telling stories about myself, how stressful it had been for “me,” a mere future magna, to be in the same classroom as “that guy.”

When Einstein Walked With Gödel: Excursions to the Edge of Thought
by Jim Holt
Published 14 May 2018

Bell, Men of Mathematics (repr., Touchstone, 1986). 7. THE AVATARS OF HIGHER MATHEMATICS G. H. Hardy, A Mathematician’s Apology (Cambridge, 1940). Michael Harris, Mathematics Without Apologies: Portrait of a Problematic Vocation (Princeton, 2015). 8. BENOIT MANDELBROT AND THE DISCOVERY OF FRACTALS Benoit Mandelbrot, The Fractalist: Memoir of a Scientific Maverick (Pantheon, 2012). Benoit Mandelbrot and Richard L. Hudson, The (Mis)behavior of Markets: A Fractal View of Financial Turbulence (Basic, 2006). 9. GEOMETRICAL CREATURES Edwin A. Abbott, The Annotated Flatland: A Romance of Many Dimensions, with an introduction and notes by Ian Stewart (Perseus, 2002).

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Indeed, the discovery of such an inconsistency would be fatal to pure mathematics, at least as we know it today. The distinction between truth and falsehood would be breached, the ladder of avatars would come crashing down, and the One Big Theorem would take a truly terrible form: 0 = 1. Yet, oddly enough, e-commerce and financial derivatives would be left untouched. 8 Benoit Mandelbrot and the Discovery of Fractals Benoit Mandelbrot, the brilliant Polish-French-American mathematician who died in 2010, had a poet’s taste for complexity and strangeness. His genius for noticing deep links among far-flung phenomena led him to create a new branch of geometry, one that has deepened our understanding of both natural forms and patterns of human behavior.

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Simon Blackburn, Truth: A Guide (Oxford, 2005). Richard Rorty, Truth and Progress: Philosophical Papers, vol. 3 (Cambridge, 1998). Acknowledgments The longer essays in this volume previously appeared, in somewhat different form, in the following publications: “A Mathematical Romance,” “The Avatars of Higher Mathematics,” “Benoit Mandelbrot and the Discovery of Fractals,” “Geometrical Creatures,” “A Comedy of Colors,” “The Dangerous Idea of the Infinitesimal,” “Dr. Strangelove Makes a Thinking Machine,” and “Einstein, ‘Spooky Action,’ and the Reality of Space” in The New York Review of Books; “When Einstein Walked with Gödel,” “Numbers Guy: The Neuroscience of Math,” “Sir Francis Galton, the Father of Statistics … and Eugenics,” “Infinite Visions: Georg Cantor v.

The Physics of Wall Street: A Brief History of Predicting the Unpredictable
by James Owen Weatherall
Published 2 Jan 2013

Given the importance that ideas like his have today, one is left to conclude that Bachelier was simply too far ahead of his time. Soon after his death, though, his ideas reappeared in the work of Samuelson and his students, but also in the work of others who, like Bachelier, had come to economics from other fields, such as the mathematician Benoît Mandelbrot and the astrophysicist M.F.M. Osborne. Change was afoot in both the academic and financial worlds that would bring these later prophets the kind of recognition that Bachelier never enjoyed while he was alive. 2 Swimming Upstream MAURY OSBORNE’S MOTHER, AMY OSBORNE, WAS AN AVID GARDENER.

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When his mentor, Jacques Hadamard, one of the most famous mathematicians of the late nineteenth century, retired from his position at the prestigious Collège de France, the Collège invited Mandelbrojt to replace him. He was a serious man, doing serious work. Or at least he would have been doing serious work if his nephew hadn’t been constantly hounding him. In 1950, Benoît Mandelbrot was a doctoral student at the University of Paris, Szolem’s alma mater, seeking (Szolem imagined) to follow in his eminent uncle’s footsteps. When Szolem first learned that Benoît wanted to pursue mathematics, he was thrilled. But gradually, Szolem began to question Benoît’s seriousness. Despite his uncle’s advice, Benoît showed no interest in the pressing mathematical matters of the day.

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He often found himself at the fringes of respectability: esteemed, though never as highly as he deserved; criticized and dismissed as much for his style as for the unconventionality of his work. Yet over the past four decades, as Wall Street and the scientific community have encountered new, seemingly insurmountable challenges, Mandelbrot’s insights into randomness have seemed ever more prescient — and more essential to understand. Benoît Mandelbrot was born in 1924, to Lithuanian parents living in Warsaw, Poland. Although his father was a businessman, two of his uncles (including Szolem) were scholars. Many of his father’s other relatives were, in Mandelbrot’s words, “wise men” with no particular employment, but with a group of followers in the community who would trade money or goods in exchange for advice or learning.

The Doomsday Calculation: How an Equation That Predicts the Future Is Transforming Everything We Know About Life and the Universe
by William Poundstone
Published 3 Jun 2019

To the right of Mitchell, though easily missed, is the familiar face of Albert Einstein, shown in profile. The speeding rocket and slow-growing hemlock allude to Einstein’s thought experiments of racing trains and light beams, used to develop his theory of relativity. Standing in front of Einstein is Benoit Mandelbrot, the IBM mathematician who described the concept of fractals. The hemlock tree and rocket blast are fractals, complex shapes in which each part resembles the whole. Zeno of Elea, a Greek philosopher whose features are known from ancient busts, dangles a cigarette. Zeno propounded the paradox of Achilles and the Tortoise.

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“According to a law established and promulgated by bald-headed, cigar-chomping know-it-alls who foregather every night at [New York deli] Lindy’s… the life expectancy of a television comedian is proportional to the total amount of his exposure on the medium.” Many comics who score a Tonight Show shot are soon forgotten, but it’s safe to assume that Jerry Seinfeld will be around awhile. Mathematician Benoit Mandelbrot came across Lindy’s law and wrote about it, saying that it applies to many things other than show business. That was Gott’s point. Before I heard of the Copernican method, I formulated a semiserious law for waiting on hold to speak to a customer support agent. Your future wait to speak to a live human is approximately equal to however long you’ve already waited.

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The reason, says Caves, is scale invariance. We need to be dealing with a process that has no characteristic time scale or lifespan, or at any rate, none that we know about. Fractals and Scale Invariance “Scale invariance” may be an unfamiliar term. Here’s one more likely to ring a bell: “fractal.” That word was coined by Benoit Mandelbrot to describe the fascinating unruliness of nature. Coastlines, snowflakes, clouds, and landscapes resist the straitjackets of Euclidean geometry. A coastline is not a “line.” A snowflake is not a hexagon. The defining quality of a fractal is scale invariance, or self-similarity. When a picture or diagram or chart of a fractal is zoomed in or out, its crinkly detail looks pretty much the same.

Think Twice: Harnessing the Power of Counterintuition
by Michael J. Mauboussin
Published 6 Nov 2012

Taleb, The Black Swan, discusses a similar concept he calls the “ludic fallacy.” 15. Donald MacKenzie, An Engine, Not a Camera: How Financial Models Shape Markets (Cambridge: MIT Press, 2006). 16. Benoit Mandelbrot, “The Variation of Certain Speculative Prices,” in The Random Character of Stock Market Prices, ed. Paul H. Cootner, (Cambridge: MIT Press, 1964), 369–412. This is also a core theme of Taleb, The Black Swan. See also Benoit Mandelbrot and Richard L. Hudson, The (Mis)Behavior of Markets (New York: Basic Books, 2004). 17. Paul H. Cootner, “Comments on The Variation of Certain Speculative Prices,” in Cootner, The Random Character of Stock Market Prices, 413–418. 18.

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If you have ever heard a financial expert refer to the stock market using terms like alpha, beta, or standard deviation, you have witnessed reductive bias in action. Most economists characterize markets using simpler, but wrong, price-change distributions. A number of high-profile financial blowups, including Long-Term Capital Management, show the danger of this bias.15 Benoit Mandelbrot, a French mathematician and the father of fractal geometry, was one of the earliest and most vocal critics of using normal distributions to explain how asset prices move.16 His chapter in The Random Character of Stock Market Prices, published in 1964, created a stir because it demonstrated that asset price changes were much more extreme than previous models had assumed.

Turing's Cathedral
by George Dyson
Published 6 Mar 2012

Archivists Christine Di Bella, Erica Mosner, and all the staff at the Institute, especially Linda Cooper, helped in every capacity, and the current trustees, especially Jeffrey Bezos, have lent continuing encouragement and support. Many of the surviving eyewitnesses—including Alice Bigelow, Julian Bigelow, Andrew and Kathleen Booth, Raoul Bott, Martin and Virginia Davis, Akrevoe Kondopria Emmanouilides, Gerald and Thelma Estrin, Benoît Mandelbrot, Harris Mayer, Jack Rosenberg, Atle Selberg, Joseph and Margaret Smagorinsky, Françoise Ulam, Nicholas Vonneumann, Willis Ware, and Marina von Neumann Whitman—took time to speak with me. “You’re within about five years of not having a testifiable witness,” Joseph Smagorinsky warned me in 2004.

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Verena Huber-Dyson (1923–): Swiss American logician and group theorist; arrived at the IAS as a postdoctoral fellow in 1948. James Brown Horner (Desmond) Kuper (1909–1992): American physicist and second husband of Mariette (Kovesi) von Neumann. Herbert H. Maass (1878–1957): Attorney and founding trustee of the IAS. Benoît Mandelbrot (1924–2010): Polish-born French American mathematician; invited by von Neumann to the IAS to study word frequency distributions in 1953. John W. Mauchly (1907–1980): American physicist, electrical engineer, and cofounder of the ENIAC project. Harris Mayer (1921–): American Manhattan Project physicist and collaborator with Edward Teller and John von Neumann.

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They tended more to think of it as a lifetime fellowship for themselves.” Veblen, adds Montgomery, “said he and Einstein and Weyl didn’t feel up to that.”49 The other Institute was the annually changing group of mostly young visitors at the beginning of their careers, interspersed with occasional established scholars taking a year off. Benoît Mandelbrot, who arrived at von Neumann’s invitation in the fall of 1953 to begin a study of word frequency distributions (sampling the occurrence of probably, sex, and Africa) that would lead to the field known as fractals, notes that the Institute “had a clear purpose and a rather strange structure in which to assemble people: heavenly bodies in residence, and then nobody, nobody, nobody, and then mostly young people.

Capitalism 4.0: The Birth of a New Economy in the Aftermath of Crisis
by Anatole Kaletsky
Published 22 Jun 2010

But in contrast to other cyclical theories that suggest that financial markets are intrinsically unstable, behavioral finance treats trend-following behavior as a temporary, and perhaps avoidable, aberration. The behavioral view is therefore less challenging to the fundamental assumption of textbook economics that markets are, on average, driven by rational calculation and are always self-stabilizing in the long term. Chaos theory was developed in the 1960s by Benoit Mandelbrot, one of the leading mathematicians of the twentieth century. Mandelbrot spent thirty years demonstrating that this theory, which transformed the study of biology, meteorology, geology, and other complex systems, could be applied also to financial markets. Mandelbrot’s research program undermined most of the mathematical assumptions of modern portfolio theory, which is the basis for the conventional risk models used by regulators, credit-rating agencies, and unsophisticated financial institutions.

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Such work has produced impressive results on industrial organization that are widely divergent from conventional economics, but these ideas have never been integrated into the study of macroeconomic policy and financial markets, where new ideas are most needed because conventional economics has clearly failed. Benoit Mandelbrot, one of the most creative mathematicians of the twentieth century and a founder of the theories of chaos and complex systems, devoted a large part of his career to studying economics and financial markets. Many of the mathematical ideas that Mandelbrot developed and that found fruitful applications in the study of earthquakes, weather, galaxies, and biological systems from the 1960s onward were inspired by his studies of finance and economics—and could be applied to these subjects with great effect.

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For more details, see Chapter 11. 4 This accelerator-multiplier concept, first proposed by Sir Roy Harrod, was later refined by Paul Samuelson and Sir John Hicks and became the standard Keynesian business cycle model. 5 Justin Lahart, “In Time of Tumult, Obscure Economist Gains Currency,” Wall Street Journal, August 18, 2007. 6 George Soros, The Soros Lectures: At the Central European University. 7 Alan Greenspan, “The Challenge of Central Banking,” remarks at the Annual Dinner and Francis Boyer Lecture of the American Enterprise Institute for Public Policy Research, Washington, DC, December 5, 1996. Available from http://www.federalreserve.gov/boarddocs/speeches/1996/19961205.htm. 8 Robert Shiller, Irrational Exuberance. 9 Benoit Mandelbrot and Richard Hudson, The (Mis)behavior of Markets: A Fractal View of Risk, Ruin and Reward, 4. 10 Nassim Nicholas Taleb, Fooled by Randomness: The Hidden Role of Chance in the Markets and in Life and the Black Swan: The Impact of the Highly Probable. 11 The term normal distribution describes prices or any other form of data that cluster predictably and reliably around a mean value in a bell curve pattern. 12 Malcolm C.

How Markets Fail: The Logic of Economic Calamities
by John Cassidy
Published 10 Nov 2009

The odds that he will get six heads in a row are one in sixty-four. The coin-tossing view of finance that Bachelier pioneered today goes under the name of the “random walk” theory, because it implies that the prices of stocks and other speculative assets will wander about aimlessly like an inebriated person. Benoit Mandelbrot, another eminent French mathematician, described the theory this way: “Suppose you see a blind drunk staggering across an open field. If you pass by again later on, how far will he have gotten? Well, he could go two steps left, three right, four backwards, and so on in an aimless jagged path.

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For the market to work at all, there must be some level of inefficiency! Grossman and Stiglitz entitled their paper “On the Impossibility of Informationally Efficient Markets.” Other economic theorists admired its terse logic, but it didn’t have much immediate impact on Wall Street. The aforementioned Benoit Mandelbrot, who is perhaps best known as one of the founders of chaos theory, was another skeptic of the efficient market hypothesis. In the early 1960s, when he was working in the research department at IBM, Mandelbrot got interested in some of the new theories that were being developed to explain how financial markets worked, and he started to gather evidence on how they performed.

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Researchers showed that stocks did better in January than in other months, and did better on Mondays than on other days of the week. They also showed that small cap stocks outperform large cap stocks; and that value stocks—those with a low price-to-dividend ratio or price-to-earnings ratio—outperform growth stocks. Confirming the point Benoit Mandelbrot made as early as 1963, researchers also demonstrated that successive movements in the market are correlated. Upward moves tend to come in clumps, and so do downward moves. And it isn’t just price changes that display this pattern: trading volumes and volatility are clustered, too. Fama himself coauthored two revisionist papers.

Currency Wars: The Making of the Next Gobal Crisis
by James Rickards
Published 10 Nov 2011

The extended analysis that follows, including elements of diversity, connectedness, interdependence and adaptability, draws on a series of lectures under the title “Understanding Complexity,” delivered in 2009 by Professor Scott E. Page of the University of Michigan. 207 However, there is strong empirical evidence, first reported by Benoît Mandelbrot . . . This discussion of fractal dimensions in market prices draws on Benoît Mandelbrot and Richard L. Hudson, The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward, New York: Basic Books, 2004. 218 Chaisson posits that the universe is best understood . . . The discussion of Chaisson’s theory of free energy rate densities is from Eric J.

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The academic counterattack on these tenets of financial economics have come from two directions. From the fields of psychology, sociology and biology came a flood of studies showing that investors are irrational after all, at least from the perspective of wealth maximization. From iconoclastic mathematical genius Benoît Mandelbrot came insights that showed future prices are not independent of the past—that the market had a kind of “memory” that could cause it to react or overreact in disruptive ways, giving rise to alternating periods of boom and bust. Daniel Kahneman and his colleague Amos Tversky demonstrated in a series of simple but brilliantly constructed experiments that individuals were full of irrational biases.

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The peaks and valleys, “double tops,” “head and shoulders” and other technical chart patterns are examples of emergence from the complexity of the overall system. Phase transitions—rapid extreme changes—are present in the form of market bubbles and crashes. Much of the work on capital markets as complex systems is still theoretical. However, there is strong empirical evidence, first reported by Benoît Mandelbrot, that the magnitude and frequency of certain market prices plot out as a power-law degree distribution. Mandelbrot showed that a time series chart of these price moves exhibited what he called a “fractal dimension.” A fractal dimension is a dimension greater than one and less than two, expressed as a fraction such as 1½; the word “fractal” is just short for “fractional.”

Antifragile: Things That Gain From Disorder
by Nassim Nicholas Taleb
Published 27 Nov 2012

So if I call someone a dangerous ethically challenged fragilista in private after the third glass of Lebanese wine (white), I will be obligated to do so here. Calling people and institutions fraudulent in print when they are not (yet) called so by others carries a cost, but is too small to be a deterrent. After the mathematical scientist Benoît Mandelbrot read the galleys of The Black Swan, a book dedicated to him, he called me and quietly said: “In what language should I say ‘good luck’ to you?” I did not need any luck, it turned out; I was antifragile to all manner of attacks: the more attacks I got from the Central Fragilista Delegation, the more my message spread as it drove people to examine my arguments.

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How does this layering operate? A tree has many branches, and these look like small trees; further, these large branches have many more smaller branches that sort of look like even smaller trees. This is a manifestation of what is called fractal self-similarity, a vision by the mathematician Benoît Mandelbrot. There is a similar hierarchy in things and we just see the top layer from the outside. The cell has a population of intercellular molecules; in turn the organism has a population of cells, and the species has a population of organisms. A strengthening mechanism for the species comes at the expense of some organisms; in turn the organism strengthens at the expense of some cells, all the way down and all the way up as well.

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For one example of a trick for debunking causality: I am not even dead yet, but am already seeing distortions about my work. Authors theorize about some ancestry of my ideas, as if people read books then developed ideas, not wondering whether perhaps it is the other way around; people look for books that support their mental program. So one journalist (Anatole Kaletsky) saw the influence of Benoît Mandelbrot on my book Fooled by Randomness, published in 2001 when I did not know who Mandelbrot was. It is simple: the journalist noticed similarities of thought in one type of domain, and seniority of age, and immediately drew the false inference. He did not consider that like-minded people are inclined to hang together and that such intellectual similarity caused the relationship rather than the reverse.

More Than You Know: Finding Financial Wisdom in Unconventional Places (Updated and Expanded)
by Michael J. Mauboussin
Published 1 Jan 2006

However, a run of thirty provides a $1.1 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. 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 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 monthly returns).

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If the largest city, Madrid, has 3 million inhabitants, the second-largest city, Barcelona, has one-half as many, the third-largest city, Valencia, one-third as many, and so forth. Zipf’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 a constant to the rank. This changes the sequence to 1/(1 + constant), 1/(2 + constant), 1/(3 + constant), etc. The second modification is to add a constant to the power of 1 in the denominator.

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The pictures will only get better with time. • Statistical properties of markets—from description to prediction? When describing markets, financial economists generally assume a definable 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 distributed along a bell-shaped curve but rather follow a power law.1 Practitioners acknowledged this fact long ago and have modified their models—even if through intuition—to accommodate this reality.

The Man From the Future: The Visionary Life of John Von Neumann
by Ananyo Bhattacharya
Published 6 Oct 2021

Von Neumann receives his Medal of Freedom from President Eisenhower. Cancer had come at a particularly cruel time. The truth was that von Neumann had been unhappy at the IAS for several years before his death. ‘Von Neumann, when I was there at Princeton, was under extreme pressure,’ says Benoît Mandelbrot, who had come to the IAS in 1953 at von Neumann’s invitation, ‘from mathematicians, who were despising him for no longer being a mathematician; by the physicists, who were despising him for never having been a real physicist; and by everybody for having brought to Princeton this collection of low-class individuals called “programmers”’.

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‘He was always gentle, always kind, always penetrating and always magnificently lucid.’3 Shy of revealing too much of himself, his good deeds were quietly done behind people’s backs. When a Hungarian-speaking factory worker in Tennessee wrote to him in 1939 asking how he could learn secondary school mathematics, von Neumann asked his friend Ortvay to send school books.4 Benoît Mandelbrot, whose stay at the IAS had been sponsored by von Neumann, unexpectedly found himself in his debt again many years later. Sometime after von Neumann’s death, prompted by a clash of personalities with his manager at IBM, Mandelbrot went looking for a new job – and found that the way had been made easier for him.

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Stan isław Ulam,’John von Neumann 1903–1957’, Bulletin of the American Mathematical Society, 64(3) (1958), pp. 1–49. 100. John von Neumann, Documentary Mathematical Association of America, 1966. Many thanks to David Hoffman, the film’s producer, for sending me the DVD in 2019. Now available to watch here: https://archive.org/details/JohnVonNeumannY2jiQXI6nrE. 101. Macrae, John von Neumann. 102. ‘Benoît Mandelbrot – Post-doctoral Studies: Weiner and Von Neumann (36/144)’, Web of Stories – Life Stories of Remarkable People, https://www.youtube.com/watch?v=U9kw6Reml6s. 103. https://rjlipton.wpcomstaging.com/the-gdel-letter/. Also see Richard J. Lipton, 2010, The P=NP Question and Gödel’s Lost Letter, Springer, New York. 104.

Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies
by Geoffrey West
Published 15 May 2017

Novel ideas that evoked concepts of discontinuities and crinkliness, which are implicit in the modern concept of fractals, were viewed as fascinating formal extensions of academic mathematics but were not generally perceived as playing any significant role in the real world. It fell to the French mathematician Benoit Mandelbrot to make the crucial insight that, quite to the contrary, crinkliness, discontinuity, roughness, and self-similarity—in a word, fractality—are, in fact, ubiquitous features of the complex world we live in.17 In retrospect it is quite astonishing that this insight had eluded the greatest mathematicians, physicists, and philosophers for more than two thousand years.

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Once again we see that underlying the daunting complexity of the natural world lies a surprising simplicity, regularity, and unity when viewed through the coarse-grained lens of scale. Although Richardson discovered this strange, revolutionary, nonintuitive behavior in his investigations of borders and coastlines and understood its origins, he didn’t fully appreciate its extraordinary generality and far-reaching implications. This bigger insight fell to Benoit Mandelbrot. Measuring the lengths of coastline using different resolutions (Britain in the example). (13) The lengths increase systematically with resolution following a power law as indicated by the examples in the graph. (14) The slope gives the fractal dimension for the coastline: the more squiggly it is, the steeper the slope.

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His paper, published in 1961, carries the marvelously obscure title “The Problem of Contiguity: An Appendix to Statistics of Deadly Quarrels,” barely revealing, even to the cognoscenti, what the content might be. Who was to know that this was to herald a paradigm shift of major significance? Well, Benoit Mandelbrot did. He deserves great credit not only for resurrecting Richardson’s work but for recognizing its deeper significance. In 1967 he published a paper in the high-profile journal Science with the more transparent title “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension.”22 This brought Richardson’s work to light by expanding on his findings and generalizing the idea.

The Misbehavior of Markets: A Fractal View of Financial Turbulence
by Benoit Mandelbrot and Richard L. Hudson
Published 7 Mar 2006

She helped review and research portions of the book; patiently transcribed many hours of tape-recorded discussions between the authors; and provided—as ever—her generous encouragement and wise companionship. For the art, we thank M. Gruskin, H. Kanzer, and M. Logan. PRELUDE by Richard L. Hudson Introducing a Maverick in Science INDEPENDENCE IS A GREAT VIRTUE. To illustrate that, Benoit Mandelbrot relates how, during the German occupation of France in World War II, his father escaped death. One day, a band of Resistance fighters attacked the prison camp where he was being held. They disarmed the guards and told the inmates to flee before the main German force struck back. So the surprised and disoriented prisoners set off towards nearby Limoges, en masse and on the high road.

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Physica A 313: 238-251. Buffett, Warren E. 1988. To the Shareholders of Berkshire Hathaway Inc. Annual Report. Omaha, Neb.: Berkshire Hathaway Inc. Burton, Jonathan. 1998. Revisiting the capital asset pricing model. Dow Jones Asset Manager May-June: 20-28. Calvet, Laurent, Adlai Fisher, and Benoit Mandelbrot. 1997. Large deviations and the distribution of price changes. Cowles Foundation Discussion Paper 1165 (September). Calvet, Laurent and Adlai Fisher. 2002. Multifractality in asset returns: Theory and evidence. Review of Economics and Statistics 84 (3): 381-406. Campbell, John Y., Andrew W.

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Multifractality: theory and evidence. An application to the French stock market. Economics Bulletin 3 (31): 1-12. Financial Executives Research Foundation. 2003. Valuing Employee Stock Options: A Comparison of 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: Making a New Science.

Chaos Kings: How Wall Street Traders Make Billions in the New Age of Crisis
by Scott Patterson
Published 5 Jun 2023

Someone told her the exotic-looking bearded gentleman in the studio had written a book about randomness. Curious, she approached him. When she discovered he was a trader, she declared she was against the market system and stomped away as Taleb sputtered in confusion. Later that day, Taleb met a man he’d soon come to idolize: Benoit Mandelbrot. The maverick French mathematician, inventor of fractal geometry, and pioneer of chaos theory, was giving a lecture at NYU’s Courant Institute about two seemingly disconnected topics—fractals and finance. Taleb was intrigued. He had no idea how finance could have anything to do with fractals.

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The French physicist was claiming to have unearthed a phantom. A phenomenon that, according to prevailing economic and financial theory, couldn’t exist. The market, according to this theory, behaves like a random walk. It was the theory first proposed in 1900 by Bachelier, the neurotic French mathematician described by Benoit Mandelbrot at NYU. Sometimes called a drunkard’s walk, the theory claims that markets—all markets—are completely random and therefore unpredictable. Imagine a drunk staggering away from a light pole. Each stagger goes in a different direction, sometimes toward the pole and sometimes away from it. By the math, it’s impossible to predict how far from the pole he’ll be by the end of the night.

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Black Swans lurk in the shadows, he warned—be careful! In the spring of 2007, The Black Swan: The Impact of the Highly Improbable hit bookstores across the U.S. It was an instant hit, debuting on the New York Times bestseller list at number five. An entire chapter, “The Aesthetics of Randomness,” was a paean to Benoit Mandelbrot, the French mathematician whose fractal geometry deeply informed Taleb’s vision of extreme events and fat tails—a land Taleb dubbed “Extremistan.” The mundane territory in the middle of the bell curve he called “Mediocristan”—the tame Gaussian world Mandelbrot had shown mostly doesn’t apply to the wild rock-’n’-roll nature of financial markets.

How Big Things Get Done: The Surprising Factors Behind Every Successful Project, From Home Renovations to Space Exploration
by Bent Flyvbjerg and Dan Gardner
Published 16 Feb 2023

The technical term for this property is “scale free,” meaning that the thing is basically the same no matter what size it is. This gives you the magic of what I call “scale-free scalability,” meaning you can scale up or down following the same principles independently of where you are 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 the pattern and discover that it looks the same as the pattern as a whole, and you keep zooming in and keep discovering the same pattern.11 Modularity can do astonishing things.

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ACKNOWLEDGMENTS Writing a book is a “Big Thing.” As such, it takes teamwork. I wish to thank the many people who made the book possible. It’s a big team, 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. Kahneman and Taleb accepted positions as distinguished research scholars with my group at Oxford, for which I cannot thank them enough.

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Probability distributions with a kurtosis lower than 3 have thinner tails than the Gaussian, which is itself considered thin-tailed. Probability distributions with a kurtosis larger than 3 are considered fat-tailed. The 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—14.5 times more fat-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.

Overcomplicated: Technology at the Limits of Comprehension
by Samuel Arbesman
Published 18 Jul 2016

You can’t have a futuristic starship that is all angles and smooth sides; you need to add ports and vents and sundry other impenetrable doodads and whatsits, pipes and bumps, indentations and grooves. Think of the ships in Battlestar Galactica or Star Wars. They are more visually intriguing thanks to their complications of unknown purpose. This process of greebling is closely related to a well-known quote from the mathematician Benoit Mandelbrot, who coined the term “fractal”: “Why is geometry often described as ‘cold’ and ‘dry’? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

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Corky Ramirez: Note that in the episode “The Van Buren Boys,” someone is referred to as “Ramirez” in a bar (though I believe his name is stressed differently than Kramer’s pronunciation of Corky Ramirez). Perhaps he is visible in the room, but it is unclear. Seinfeld superfans: please send me mail. delightfully evocative term: “greeblies”: : Or, alternatively, “greebles.” Kelly, What Technology Wants, 318. the mathematician Benoit Mandelbrot: Benoit B. Mandelbrot, The Fractal Geometry of Nature (New York: W. H. Freeman and Company, 1982), 1. Recall “Funes the Memorious”: Borges, “Funes, His Memory,” in Collected Fictions, 131–37. “The patterns of a river network”: Philip Ball, Branches, vol. 3 of Nature’s Patterns: A Tapestry in Three Parts (Oxford, UK: Oxford University Press, 2009), 181.

This Will Make You Smarter: 150 New Scientific Concepts to Improve Your Thinking
by John Brockman
Published 14 Feb 2012

You can see their optimism (or anxiety) about how technology is changing culture and interaction. You’ll observe a frequent desire to move beyond deductive reasoning and come up with more rigorous modes of holistic or emergent thinking. You’ll also get a sense of the emotional temper of the group. People in this culture love neat puzzles and cool questions. Benoit Mandelbrot asked his famous question “How long is the coast of Britain?” long before this symposium was written, but it perfectly captures the sort of puzzle people in this crowd love. The question seems simple. Just look it up in the encyclopedia. But as Mandelbrot observed, the length of the coast of Britain depends on what you use to measure it.

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The hardest problem in software is controlling the tendency of software systems to grow incomprehensibly complex. Recursive structure helps convert impenetrable software rain forests into French gardens—still (potentially) vast and complicated but much easier to traverse and understand than a jungle. Benoit Mandelbrot famously recognized that some parts of nature show recursive structure of a sort: A typical coastline shows the same shape or pattern whether you look from six inches or sixty feet or six miles away. But it also happens that recursive structure is fundamental to the history of architecture, especially to the Gothic, Renaissance, and Baroque architecture of Europe—covering roughly the five hundred years between the thirteenth and eighteenth centuries.

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Lately, one of many projects has been to revisit the aesthetic space of scientific visualizations, and another the epitome of mathematics made tangible: fractals, which I had done almost twenty years ago with virtuoso coder Ben Weiss, now enjoying them via realtime flythroughs on a handheld little smartphone. Here was the most extreme example: A tiny formula, barely one line on paper, used recursively, yields worlds of complex images of amazing beauty. (Ben had the distinct pleasure of showing Benoit Mandelbrot an alpha version at a TED conference just months before Mandelbrot’s death.) My hesitation about overuse of parsimony was expressed perfectly in a quote from Albert Einstein, arguably the counterpart blade to Ockham’s razor: “Things should be made as simple as possible—but not simpler.” And there we have the perfect application of its truth, used recursively on itself: Neither Einstein nor Ockham actually used the exact words as quoted!

The Bed of Procrustes: Philosophical and Practical Aphorisms
by Nassim Nicholas Taleb
Published 30 Nov 2010

* Moore’s Law stipulates that computational power doubles every eighteen months. * Say, Sarah Palin. † The biggest error since Socrates has been to believe that lack of clarity is the source of all our ills, not the result of them. AESTHETICS Art is a one-sided conversation with the unobserved. – The genius of Benoît Mandelbrot is in achieving aesthetic simplicity without having recourse to smoothness. – Beauty is enhanced by unashamed irregularities; magnificence by a façade of blunder. – To understand “progress”: all places we call ugly are both man-made and modern (Newark), never natural or historical (Rome)

The Joys of Compounding: The Passionate Pursuit of Lifelong Learning, Revised and Updated
by Gautam Baid
Published 1 Jun 2020

According to Sebastian Mallaby, In the early 1960s, a maverick mathematician named Benoit Mandelbrot argued that the tails of the distribution might be fatter than the normal bell curve assumed; and Eugene Fama, the father of efficient-market theory, who got to know Mandelbrot at the time, conducted tests on stock-price changes that confirmed Mandelbrot’s assertion. If price changes had been normally distributed, jumps greater than five standard deviations should have shown up in a daily price data about once every seven thousand years. Instead, they cropped up about once every three to four years [emphasis added].9 Benoit Mandelbrot was a Polish-born mathematician and polymath who developed a new branch of mathematics known as fractal geometry, which recognizes the hidden order in the seemingly disordered, the plan in the unplanned, the regular pattern in the irregularity of nature.

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Charles Mackay, Charles Kindleberger, John Galbraith, John Brooks, Edward Chancellor, Robert Shiller, and Maggie Mahar educated me on the history of market cycles, speculative manias, and the subsequent busts. Peter Senge and Donella Meadows educated me on systems thinking and a more interconnected view of the world. George Soros, Benoit Mandelbrot, and Richard Bookstaber made me aware of the intricate and highly dynamic feedback loops present in markets and social systems. John Maynard Keynes enlightened me on the significance of prevailing sentiments in markets and economies, and the critical role of timely government intervention. Burton Malkiel, Charles Ellis, and John Bogle taught me the importance of minimizing costs and staying the course.

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Hagstrom, The Warren Buffett Way, 2nd ed. (Hoboken, NJ: Wiley, 2005). 7. Quoted in Foulke, “Warren Buffett on LTCM.” 8. Buffett FAQ, 2006 Berkshire Hathaway Annual Meeting, http://buffettfaq.com. 9. Sebastian Mallaby, More Money Than God: Hedge Funds and the Making of a New Elite (London: Penguin, 2011). 10. Benoit Mandelbrot and Richard L. Hudson, The (Mis)Behavior of Markets: A Fractal View of Financial Turbulence (New York: Basic Books, 2006), 20, 217, 248. 11. Benjamin Graham and David Dodd, Security Analysis: The Classic 1934 Edition (New York: McGraw-Hill Education, 1996). 12. Howard Marks, The Most Important Thing Illuminated: Uncommon Sense for the Thoughtful Investor (New York: Columbia University Press, 2013). 13.

Complexity: A Guided Tour
by Melanie Mitchell
Published 31 Mar 2009

If you then view the same coastline from your car on the coast highway, it still appears to have the exact same kind of ruggedness, but on a smaller scale (Figure 7.2, bottom). Ditto for the close-up view when you stand on the beach and even for the ultra close-up view of a snail as it crawls on individual rocks. The similarity of the shape of the coastline at different scales is called “self-similarity.” The term fractal was coined by the French mathematician Benoit Mandelbrot, who was one of the first people to point out that the world is full of fractals—that is, many real-world objects have a rugged self-similar structure. Coastlines, mountain ranges, snowflakes, and trees are often-cited examples. Mandelbrot even proposed that the universe is fractal-like in terms of the distribution of galaxies, clusters of galaxies, clusters of clusters, et cetera.

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Zipf himself proposed that, on the one hand, people in general operate by a “Principle of Least Effort”: once a word has been used, it takes less effort to use it again for similar meanings than to come up with a different word. On the other hand, people want language to be unambiguous, which they can accomplish by using different words for similar but nonidentical meanings. Zipf showed mathematically that these two pressures working together could produce the observed power-law distribution. In the 1950s, Benoit Mandelbrot, of fractal fame, had a somewhat different explanation, in terms of information content. Following Claude Shannon’s formulation of information theory (cf. chapter 3), Mandelbrot considered a word as a “message” being sent from a “source” who wants to maximize the amount of information while minimizing the cost of sending that information.

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Ingalls et al., Proceedings of the 2004 Winter Simulation Conference, pp. 130–141. Piscataway, NJ: IEEE Press, 2004. “This relation is now called Zipf’s law”: Zipf’s original publication on this work is a book: Zipf, G. K., Selected Studies of the Principle of Relative Frequency in Language. Cambridge, MA: Harvard University Press, 1932. “Benoit Mandelbrot … had a somewhat different explanation”: Mandelbrot. B., An informational theory of the statistical structure of languages. In W. Jackson (editor), Communicaiton Theory, Woburn, MA: Butterworth, 1953, pp. 486–502. “Herbert Simon proposed yet another explanation”: Simon, H. A., On a class of skew distribution functions.”

A Mathematician Plays the Stock Market
by John Allen Paulos
Published 1 Jan 2003

Elliott believed as well that these patterns exist at many levels and that any given wave or cycle is part of a larger one and contains within it smaller waves and cycles. (To give Elliott his due, this idea of small waves within larger ones having the same structure does seem to presage mathematician Benoit Mandelbrot’s more sophisticated notion of a fractal, to which I’ll return later.) Using Fibonacci-inspired rules, the investor buys on rising waves and sells on falling ones. The problem arises when these investors try to identify where on a wave they find themselves. They must also decide whether the larger or smaller cycle of which the wave is inevitably a part may temporarily be overriding the signal to buy or sell.

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The surface of the mountain looks roughly the same whether seen from a height of 200 feet by a giant or close up by an insect. The branching of a tree appears the same to us as it does to birds, or even to worms or fungi in the idealized limiting case of infinite branching. As the mathematician Benoit Mandelbrot, the discoverer of fractals, has famously written, “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” These and many other shapes in nature are near fractals, having characteristic zigzags, push-pulls, bump-dents at almost every size scale, greater magnification yielding similar but ever more complicated convolutions.

Near and Distant Neighbors: A New History of Soviet Intelligence
by Jonathan Haslam
Published 21 Sep 2015

It was the gifted son of Russian emigrés in the United States, William Friedman, who discovered the index of coincidence: the likelihood of a given letter in any text finding itself in exactly the same position in another text, even a ciphered text.39 This approach was then trumped by research into the application of statistics to linguistics pioneered by George Zipf in 193540 and more rigorously articulated in mathematical form by Benoît Mandelbrot after the war, funded by the U.S. armed services.41 Zipf found degrees of probability of a word appearing in a text; more than that, a fixed ratio of repetition between the commonest word and the next most common word, and so forth. As he wrote, words are not deliberately chosen for their frequency, but they “have a frequency distribution of great orderliness which for a large portion of the curve seems to be constant for language in general.”42 In the United States, cryptolinguistics was coming into being as a field in its own right.

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The Russians, they were relieved to discover, “had yet to perfect their cryptosecurity procedures.”53 New American comparators (such as Warlock) were already operating at speed by weighing each letter according to language frequency.54 The next step was to replicate the chi-square distribution test mechanically, to compare the frequency with which a letter appeared in one text to the frequency with which it appeared in another, and testing to ensure that this was not just a matter of chance. This had to be done through thousands of multiplications and additions—and at speed.55 Meanwhile, cryptolinguistics were developing with increasing sophistication under Benoît Mandelbrot. Though apparently produced at random with respect to the probability of repetition, words emerge in the text in a discernible pattern that bears no relationship to grammar or meaning. This means that even were a text enciphered, the probability of a word appearing remained just as it did in plain text.

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The Mitrokhin Archive: The KGB in Europe and the West. London and New York: Allen Lane, 2000. ______. The Mitrokhin Archive II: The KGB and the World. London and New York: Allen Lane, 2005. Antonov, Vladimir, and Vladimir Karpov. Tainye informatory Kremlya: Vollenberg, Artuzov i drugie. Moscow: Geia Iterum, 2001. Apostel, Léo, Benoît Mandelbrôt, and Albert Morf. Logique, Langage et Théorie de l’Information. Paris: Presses Universitaires de France, 1957. Benson, Robert. The Venona Story. Meade, MD: NSA, 2000. Bezymensky, Lev. Budapeshtskaya missiya: Raul’ Vallenberg. Moscow: Kollektsiya “Sovershenno Sekretno,” 2001. Boltunov, Mikhail.

Money Changes Everything: How Finance Made Civilization Possible
by William N. Goetzmann
Published 11 Apr 2016

We shall see that the work of three of the giants of modern finance, Robert Merton, Fischer Black, and Myron Scholes, built directly on the insights and techniques of the French mathematical tradition—both in terms of its strength and also its weakness. This last point is reserved for a discussion about a modern French mathematician, and my former Yale colleague, Benoit Mandelbrot. RANDOM WALKS Almost nothing is known about the nineteenth-century French stockbroker Jules Regnault (1834–1894). What we do know comes from the efforts of Franck Jovanovic, a lecturer in finance at Leicester University. Over the past decade, Jovanovic has studied the intellectual development of mathematical finance and traced a key logical foundation of modern quantitative methods to Jules Regnault, a successful broker on the Paris Bourse during the middle of the nineteenth century.2 In 1863, Regnault wrote a strikingly novel book, Calcul des Chances et Philosophie de la Bourse, arguing that it is impossible to profit by speculating in the market.

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In fact, the non-normality of security prices had been well known for decades prior to the crash of 2008—and for that matter the crash of 1987, as was the potential for extreme events. The “high priest” of non-normality before Nassim Taleb ever started to trade or write about extreme events was Benoit Mandelbrot, the creator of fractal geometry, a mathematician who both carried the mantle of French mathematical finance and who also believed he had discovered its fatal flaw. Mandelbrot was a student of Paul Lévy’s—the son of the man who gave Bachelier bad marks at his examination at the École Polytechnique in 1900.

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Brownian motion was just one process in the family of Lévy processes—and perhaps the best behaved of them. Other stochastic processes have such things as discontinuous jumps and unusually large shocks (which might, for example, explain the crash of 1987, when the US stock market lost 22.6% of its value in a single day). In the 1960s, Benoit Mandelbrot began to investigate whether Lévy processes described economic time series like cotton prices and stock prices. He found that the ones that generated jumps and extreme events better described financial markets. He developed a mathematics around these unusual Lévy processes that he called “fractal geometry.”

Them And Us: Politics, Greed And Inequality - Why We Need A Fair Society
by Will Hutton
Published 30 Sep 2010

There is an enormous intellectual and financial investment in the status quo. Academics have built careers, reputations and tenure on a particular view of the world being right. Only an earthquake can persuade them to put up their hands and acknowledge they were wrong. When the mathematician Benoit Mandelbrot began developing his so-called fractal mathematics and power laws in the early 1960s, arguing that the big events outside the normal distribution are the ones that need explaining and assaulting the whole edifice of mathematical theory and the random walk, MIT’s Professor Paul Cootner (the great random walk theorist) exclaimed: ‘surely, before consigning centuries of work to the ash pile, we should like some assurance that all our work is truly useless’.

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See Brad DeLong, Andrei Shleifer, Larry Summers and Michael Waldman (1990) ‘Noise Trader Risk in Financial Markets’, Journal of Political Economy 98: 703–38. 35 Anil Kashyap, Raghuram Rajan and Jeremy Stein (2008) ‘Rethinking Capital: Regulation’, paper for the Federal Reserve Bank of Kansas City. 36 Andrew Haldane (2009) ‘Why Banks Failed the Stress Test’, presentation to the Marcus-Evans Conference on Stress-Testing, 9–10 February. 37 James G. Rickards, ‘The Risks of Financial Modeling: VaR and the Economic Meltdown’, testimony before the Subcommittee on Investigations and Oversight Committee on Science and Technology, US House of Representatives, 10 September 2009. 38 Benoit Mandelbrot (2008) The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin and Reward, Profile Books. For another interesting example of cross-fertilisation, see Didier Sornette (2003) Why Stockmarkets Crash: Critical Events in Complex Financial Systems, Princeton University Press. 39 See Justin Fox (2009) The Myth of the Rational Market: A History of Risk, Reward, and Delusion on Wall Street, HarperBusiness. 40 The following example is paraphrased from Baseline Scenario: http://baselinescenario.com/2009/10/01/the-economics-of-models/. 41 Gillian Tett (2009) Fool’s Gold: How Unrestrained Greed Corrupted a Dream, Shattered Global Markets and Unleashed a Catastrophe, Little, Brown. 42 Lucien Bebchuk and Jesse Fried (2004) Pay without Performance: The Unfulfilled Promise of Executive Compensation, Harvard University Press. 43 Lucian Bebchuk and Holger Spamann (2009) ‘Regulating Bankers’ Pay’, Harvard Law and Economics Discussion Paper No. 641. 44 Jesse Eisinger, ‘London Banks, Falling Down’, Portfolio, 13 August 2008, at http://www.portfolio.com/views/columns/wall-street/2008/08/13/Problemsin-British-Banking-System/. 45 Philip Augar (2009) Chasing Alpha: How Reckless Growth and Unchecked Ambition Ruined the City’s Golden Decade, The Bodley Head. 46 Albert-Laszlo Baraasi (2002) Linked: The New Science of Networks, Basic Books.

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See also Matthew Jackson (2008) Social and Economic Networks, Princeton University Press. 47 Nicholas Christakis and James Fowler (2010) Connected: The Amazing Power of Social Lives and How They Shape Our Lives, Harper Press. 48 Robert M. May, Simon A. Levin and George Sugihara (2008) ‘Ecology for Bankers’, Nature 451 (21): 893–5. 49 Richard Bookstaber (2007) A Demon of Our Own Design: Markets, Hedge Funds, and the Perils of Financial Innovation, John Wiley & Sons. 50 Cited by Benoit Mandelbrot (2008) The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin and Reward, Profile Books, p. 154. 51 Ibid. 52 Andrew Haldane (2009) ‘Rethinking the Financial Network’, presentation to the Financial Students Association, Amsterdam. 53 Bobbi Low, Elinor Ostrom, Carl Simon and James Wilson, ‘Redundancy and Diversity’, in Wilson Fikret Berkes, Johan Colding and Carl Folke (eds) (2003) Navigating Social-Ecological Systems: Building Resilience for Complexity and Change, Cambridge University Press. 54 Scott Page (2007) The Difference: How the Power of Diversity Creates Better Groups, Firms, Schools, and Societies, Princeton University Press.

The Success Equation: Untangling Skill and Luck in Business, Sports, and Investing
by Michael J. Mauboussin
Published 14 Jul 2012

Michael Bar-Eli, Simcha Avugos, and Markus Raab, “Twenty Years of ‘Hot Hand’ Research: Review and Critique,” Psychology of Sport and Exercise 7, no. 6 (November 2006): 525–553; and Alan Reifman, Hot Hands: The Statistics Behind Sports' Greatest Streaks (Washington, DC: Potomac Books, 2011). 7. Frank H. Knight, Risk, Uncertainty, and Profit (New York: Houghton and Mifflin, 1921), and http://www.econlib.org/library/Knight/knRUP.html. Benoit Mandelbrot distinguishes between “mild” and “wild” chance. These terms neatly capture the spirit of this discussion; see Benoit Mandelbrot and Richard L. Hudson, The (Mis)Behavior of Markets (New York: Basic Books, 2004), 32–33. 8. William Goldman, Adventures in the Screen Trade: A Personal View of Hollywood and Screenwriting (New York: Warner Books, 1983), 39. 9.

How Not to Network a Nation: The Uneasy History of the Soviet Internet (Information Policy)
by Benjamin Peters
Published 2 Jun 2016

In 1947, the year before he published Cybernetics with the MIT Press, Wiener attended Szolem Mandelbrot’s congress on harmonic analysis in Nancy, France, which resulted in a French book contract for the book that, while initially resisted by the MIT Press, sold a sensational 21,000 copies over three reprints in six months after its release in 1948. Three years later, in 1951, at the invitation of Benoit Mandelbrot, the founder of fractals and Szolem’s nephew, Wiener returned to lecture at Collège de France. Between 1947 and 1952, a flurry of press coverage and public controversy sprung up between two camps of anticybernetic communists and anticommunist cyberneticists.32 (Jacques Lacan, who served in the French army, may very well have been among the anticommunists and early cyberneticists at the time.)

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Aleksandr Bogdanov—old Bolshevik revolutionary, right-hand man to Vladimir Lenin, and philosopher—developed a wholesale theory that analogized between society and political economy, which he published in 1913 as Tektology: A Universal Organizational Science, a proto-cybernetics minus the mathematics, whose work Wiener may have seen in translation in the 1920s or 1930s.39 Stefan Odobleja was a largely ignored Romanian whose pre–World War II work prefaced cybernetic thought.40 John von Neumann, the architect of the modern computer, a founding game theorist, and a Macy Conference participant, was a Hungarian émigré. Szolem Mandelbrojt, a Jewish Polish scientist and uncle of fractal founder Benoit Mandelbrot, organized Wiener’s collaboration on harmonic analysis and Brownian motion in 1950 in Nancy, France. Roman Jakobson, the aforementioned structural linguist, a collaborator in the Macy Conferences, and a Russian émigré, held the chair in Slavic studies at Harvard founded by Norbert Wiener’s father.

Licence to be Bad
by Jonathan Aldred
Published 5 Jun 2019

If there was an answer, it would surely be revealed by the physical properties of snowflakes. But if we look at snowflakes under a magnifying lens we find a different kind of property. Snowflakes are called ‘scale-invariant’ by physicists because their crystal structure looks the same no matter how much we magnify them. Snowflakes are an example of what the mathematician Benoît Mandelbrot calls fractals – structures with no natural or normal size and which recur at different scales. (Another example is trees: the pattern of branches looks like the pattern of leaves on a branch, and also the pattern of veins in a leaf). Mandelbrot noticed that prices in financial markets have this property: a graph showing the price over time of some stock or market index will look much the same, whether the time period covered is several decades, a few seconds, or anything in between.

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The late Tony Atkinson was a world authority on inequality and this is the best recent book for a careful and thorough discussion of the facts of inequality. 2 Mishel, L., and Sabadish, N., CEO Pay in 2012 was Extraordinarily High Relative to Typical Workers and Other High Earners (Economic Policy Institute, 2013). 3 Speech at the Royal Geographical Society Presidential Dinner, London, 1991. 4 On histories of the effect of Reagan’s and Thatcher’s ideas, one inspiration for this book was Daniel Rodgers’s superb Age of Fracture (Harvard: Harvard University Press, 2011). See especially chapter 2. 5 Strathern, P. (2001), Dr Strangelove’s Game (London: Hamish Hamilton), 227. 6 Atkinson, 19–20. 7 See Economist, 13 October 2012, ‘The Rich and the Rest’, and research cited there. 8 Quoted in Benoit Mandelbrot; Hudson, Richard L. (2004), The (Mis) behavior of Markets: A Fractal View of Risk, Ruin, and Reward (New York: Basic Books), 155. 9 Hacker, J., and Pierson, P. (2010), ‘Winner-Take-All Politics’, Politics and Society, 38 (2010), 152–204. 10 See for instance Atkinson, 80–81 and J. Stiglitz (2012), The Price of Inequality (London: Allen Lane), 27–8. 11 Norton, M., and Ariely, Dan, ‘Building a Better America – One Wealth Quintile at a Time’, Perspectives in Psychological Science, 6 (2011), 9–12; Davidai, S., and Gilovich, T. (2015), ‘Building a More Mobile America – One Income Quintile at a Time’, in ibid., 10, 60–71; survey conducted by Fondation-Jean-Jaurès at https://jean-jaures.org/nos-productions/la-perception-des-inegalites-dans-le-monde. 12 See for instance M.

Capital Ideas: The Improbable Origins of Modern Wall Street
by Peter L. Bernstein
Published 19 Jun 2005

Twenty years after writing his dissertation, he remarked that his analysis had embodied “images taken from natural phenomena . . . a strange and unexpected linkage and a starting point for great progress.” His superiors did not agree. Although Poincarè, his teacher, wrote that “M. Bachelier has evidenced an original and precise mind,” he also observed that “The topic is somewhat remote from those our candidates are in the habit of treating.”5 Benoit Mandelbrot, the pioneer of fractal geometry and one of Bachelier’s great admirers, recently suggested that no one knew where to pigeonhole Bachelier’s findings. There was no ready means to retrieve them, assuming that someone wanted to. Sixty years were to pass before anyone took the slightest notice of his work. ••• The key to Bachelier’s insight is his observation, expressed in a notably modern manner, that “contradictory opinions concerning [market] changes diverge so much that at the same instant buyers believe in a price increase and sellers believe in a price decrease.”6 Convinced that there is no basis for believing that—on the average—either sellers or buyers consistently know any more about the future than the other, he arrived at an astonishing conjecture: “It seems that the market, the aggregate of speculators, at a given instant can believe in neither a market rise nor a market fall, since, for each quoted price, there are as many buyers as sellers.”7 (emphasis added) The fond hopes of home buyers in California during the 1980s provide a vivid example of Bachelier’s perception.

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These five works formed the basis for the important and farreaching research into this question that was to emerge at MIT around 1970 and that will occupy our attention later on. Cootner’s book also contained a short article by Fama, reprinted from the Journal of Business for October 1963, in which Fama expanded on an analysis of market behavior conducted by Benoit Mandelbrot, a French mathematician living in the United States whose work was published in the same issue of the journal. Mandelbrot proposed that stock prices fluctuate so irregularly because they are not sufficiently well behaved to submit to the kind of rigorous statistical analysis recommended by Bachelier and Samuelson.

Think Complexity
by Allen B. Downey
Published 23 Feb 2012

In their 2004 paper, “Efficient algorithm for the forest fire model,” they present evidence that the system is not critical after all (http://pre.aps.org/abstract/PRE/v70/i6/e066707). How do these results bear on Bak’s claim that SOC explains the prevalence of critical phenomena in nature? Example 9-8. In The Fractal Geometry of Nature, Benoit Mandelbrot proposes what he calls a “heretical” explanation for the prevalence of long-tailed distributions in natural systems (page 344). It may not be, as Bak suggests, that many systems can generate this behavior in isolation. Instead there may be only a few, but there may be interactions between systems that cause the behavior to propagate.

Infinite Ascent: A Short History of Mathematics
by David Berlinski
Published 2 Jan 2005

Those calculations represent dog work, and mathematicians are notably inferior to the computer when it comes to going to the dogs. (Many mathematicians cannot, in fact, add a simple column of figures with the accuracy expected of a German greengrocer.) An uneasy feeling nonetheless persists that the method of proof has somehow been compromised. No one has quite said why. Benoit Mandelbrot—a distant cousin of mine, a remote family connection—is a mathematician who has immensely enriched the ordinary happiness of mankind by showing how beautiful pictures can be made simply on the computer. His images are now everywhere and are everywhere known as Mandelbrot sets. Their construction depends on recursive iteration and a computer program that can assign colors to regions of the complex plane.

On the Future: Prospects for Humanity
by Martin J. Rees
Published 14 Oct 2018

Conway indulged in a lot of ‘trial and error’ before he came up with a simple ‘virtual world’ that allowed for interesting emergent variety. He used pencil and paper, before the days of personal computers, but the implications of the Game of Life only emerged when the greater speed of computers could be harnessed. Likewise, early PCs enabled Benoit Mandelbrot and others to plot out the marvellous patterns of fractals—showing how simple mathematical formulas can encode intricate apparent complexity. Most scientists resonate with the perplexity expressed in a classic essay by the physicist Eugene Wigner, titled ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’.2 And also with Einstein’s dictum that ‘the most incomprehensible thing about the universe is that it is comprehensible’.

Radical Uncertainty: Decision-Making for an Unknowable Future
by Mervyn King and John Kay
Published 5 Mar 2020

The nineteenth of October 1987, on which the principal American stock indices fell by around 20% during the day, is the financial analogue of the Valdivia earthquake. Extreme events are common with power laws and rare in normal distributions. The application of power laws to economics was pioneered in the early 1960s by the Polish-French-American mathematician Benoit Mandelbrot. He established that movements in cotton prices could be described by a power law. 9 Power laws have a property of ‘scale invariance’. If you look at a snowflake under a powerful microscope, the shape of every small part you see is the same as the shape you see with the naked eye. The property which creates this beautiful structure is called fractal geometry.

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Lorraine Daston provides a comprehensive account of the development of probabilistic reasoning in Classical Probability in the Enlightenment (1995) and its application to insurance is described in Niall Ferguson’s The Ascent of Money (2008). In 2019 the American Statistical Association devoted an entire issue to the misuse of probabilistic reasoning to make inferences about causation. The editorial concluded ‘it is time to stop using the term “statistically significant” entirely’. * The study of power laws was pioneered by Benoit Mandelbrot, and Mark Buchanan’s Ubiquity (2002) is a survey of many applications. Again as we went to press, we saw Ian Stewart’s Do Dice Play God? (2019) which reviews several of the puzzles and paradoxes in the early chapters of this book. ____________ * Ronald L. Wasserstein, Allen L. Schirm and Nicole A.

Statistical Arbitrage: Algorithmic Trading Insights and Techniques
by Andrew Pole
Published 14 Sep 2007

As you read the foregoing description, you may feel a sense of deja vu. The description of modeling the variation about the mean during periods of zero forecast activity is quite the same as the general description of 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 currently in the mathematical finance literature.

The Wisdom of Finance: Discovering Humanity in the World of Risk and Return
by Mihir Desai
Published 22 May 2017

The original works in this stream of research are well discussed in this pioneering paper: Fama, Eugene. “Efficient Capital Markets: A Review of Theory and Empirical Work.”Journal of Finance 25, no. 2 (May 1970): 383–417. In particular, Fama is generous with his referencing of earlier work, including that of Paul Samuelson, Bill Sharpe, Benoit Mandelbrot, Paul Cootner, Jack Treynor, and others. This lecture is an excellent source on the ideas of efficient markets: Fama, Eugene. “A Brief History of the Efficient Market Hypothesis.” Lecture, Masters of Finance. February 12, 2014. https://www.youtube.com/watch?v=NUkkRdEknjI. An alternative stream of important research on this topic was triggered by Grossman, Sanford J., and Joseph E.

The Biology of Belief: Unleashing the Power of Consciousness, Matter & Miracles
by Bruce H. Lipton
Published 1 Jan 2005

For example, you cannot map a tree, a cloud, or a mountain using the mathematical formulas of this geometry. In nature, most organic and inorganic structures display more irregular and chaotic-appearing patterns. These natural images can only be created by using the recently discovered mathematics called fractal geometry. French mathematician Benoit Mandelbrot launched the field of fractal mathematics and geometry in 1975. Like quantum physics, fractal (fractional) geometry forces us to consider those irregular patterns, a quirkier world of curvy shapes and objects with more than three dimensions. The mathematics of fractals is amazingly simple because you need only one equation, using only simple multiplication and addition.

The Simpsons and Their Mathematical Secrets
by Simon Singh
Published 29 Oct 2013

When I asked him how he knew the formula would be a cubic polynomial, he said: “What else would it be?” APPENDIX 4 Fractals and Fractional Dimensions We normally think of fractals as patterns that consist of self-similar patterns at every scale. In other words, the overall pattern associated with an object persists as we zoom in and out. As the father of fractals Benoit Mandelbrot pointed out, these self-similar patterns are found in nature: “A cauliflower shows how an object can be made of many parts, each of which is like a whole, but smaller. Many plants are like that. A cloud is made of billows upon billows upon billows that look like clouds. As you come closer to a cloud you don’t get something smooth but irregularities at a smaller scale.”

Extreme Money: Masters of the Universe and the Cult of Risk
by Satyajit Das
Published 14 Oct 2011

Hearing complaints that his strategies were not working, Asness’ wife asked him incredulously: “Let me get this straight. I thought you said you make your money because people aren’t completely rational. Yet now you’re mad because they’re too irrational.”24 Risk management assumes that price changes are normally distributed. The mathematician Benoit Mandelbrot demonstrated that normal distributions do not exist in practice. In Fooled by Randomness and Black Swan, Nicholas Nassim Taleb argued against the application of statistical methods in finance, especially the normal distribution curve to measure risk. Taleb drew on John Stuart Mill, himself rephrasing a problem posed by Scottish philosopher David Hume: “no amount of observations of white swans can allow the inference that all swans are white, but the observation of a single black swan is sufficient to refute that conclusion.”

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Like the philosoher of science Thomas Kuhn, financial markets persisted with flawed models, arguing that they worked in normal conditions and were superior to alternatives. David Einhorn compared risk systems to: “an airbag which works all the time except when you get into a crash.”25 Financiers were reluctant to use qualitative approaches that were inconsistent with their scientific self-image. As Benoit Mandelbrot, the creator of chaos theory, observed: “Human Nature yearns to see order and hierarchy in the world. It will invent it if it cannot find it.”26 Psychologist B.F. Skinner created “superstitious pigeons.” Unlike in normal experiments, he fed the pigeons with no reference to the bird’s behavior.

Memory Machines: The Evolution of Hypertext
by Belinda Barnet
Published 14 Jul 2013

This hazard was apparent to the troubadour and know-hit wonder Jonathan Coulton, when he wrote one of the great tunes of popular science, ‘Mandelbrot Set’: Mandelbrot’s in heaven At least he will be when he’s dead Right now he’s still alive and teaching math at Yale The song was released in October 2004, giving it a nice run of six years before its lyrics were compromised by Benoît Mandelbrot’s passing in 2010. Even thus betrayed to history, ‘Mandelbrot Set’ still marks the contrast between extraordinary and ordinary lives, dividing those who change the world, in ways tiny or otherwise, from those who sing about them or merely ruminate. The life of ideas, perhaps like ontogeny, works through sudden transformations and upheavals, apparent impasses punctuated by instant, lateral shift.

The Education of a Value Investor: My Transformative Quest for Wealth, Wisdom, and Enlightenment
by Guy Spier
Published 8 Sep 2014

Tartakower and J. du Mont Homo Ludens: A Study of the Play Element in Culture by Johan Huizinga Reality Is Broken: Why Games Make Us Better and How They Can Change the World by Jane McGonigal Winning Chess Tactics for Juniors by Lou Hays Wise Choices: Decisions, Games, and Negotiations by Richard Zeckhauser, Ralph Keeney, and James Sebenius Investing A Zebra in Lion Country by Ralph Wanger with Everett Mattlin Active Value Investing: Making Money in Range-Bound Markets by Vitaliy Katsenelson Beating the Street by Peter Lynch Common Stocks and Uncommon Profits by Philip Fisher Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets by Nassim Nicholas Taleb Fooling Some of the People All of the Time: A Long Short Story by David Einhorn and Joel Greenblatt Fortune’s Formula: The Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street by William Poundstone Investing: The Last Liberal Art by Robert Hagstrom Investment Biker: Around the World with Jim Rogers by Jim Rogers More Mortgage Meltdown: 6 Ways to Profit in These Bad Times by Whitney Tilson and Glenn Tongue More Than You Know: Finding Financial Wisdom in Unconventional Places by Michael Mauboussin Of Permanent Value: The Story of Warren Buffett by Andrew Kilpatrick Pioneering Portfolio Management: An Unconventional Approach to Institutional Investment by David Swensen Security Analysis by Benjamin Graham and David Dodd Seeking Wisdom: From Darwin to Munger by Peter Bevelin Short Stories from the Stock Market: Uncovering Common Themes behind Falling Stocks to Find Uncommon Ideas by Amit Kumar The Dhandho Investor: The Low-Risk Value Method to High Returns by Mohnish Pabrai The Manual of Ideas: The Proven Framework for Finding the Best Value Investments by John Mihaljevic The Misbehavior of Markets: A Fractal View of Financial Turbulence by Benoit Mandelbrot and Richard Hudson The Most Important Thing: Uncommon Sense for the Thoughtful Investor by Howard Marks The Warren Buffett Way by Robert Hagstrom Value Investing: From Graham to Buffett and Beyond by Bruce Greenwald, Judd Kahn, Paul Sonkin, and Michael van Biema Where Are the Customers’ Yachts?

The Patterning Instinct: A Cultural History of Humanity's Search for Meaning
by Jeremy Lent
Published 22 May 2017

The li of a thing is one with the li of all things…. There is only one li in the world.” While this might sound rather mystical, recent breakthroughs in mathematics have demonstrated Cheng's statements to be a perceptive insight into the nature of reality. Fractal geometry, pioneered by mathematician Benoit Mandelbrot, shows how nature forms intricate patterns that replicate themselves at different scales, each pattern nested inside another. Examples of these fractal patterns are observable in clouds, coastlines, ferns, and sand dunes.41 Since Mandelbrot's discovery, biologists have come to recognize that the design of life itself is fractal, with cells self-organizing to form organisms, which then self-organize into communities of organisms and ecosystems.

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In pursuing their disciplines, scientists would often use the Latin phrase ceteribus paribus—“other things being equal”—to dismiss the random noise that didn't fit into the theory. Now, in systems thinking, a new set of methods was emerging to investigate the unequal world of those other things.20 A brilliant mathematician, Benoit Mandelbrot, developed a new branch of mathematics, called fractal geometry, to describe this non-Newtonian world. His 1983 book The Fractal Geometry of Nature had a profound effect on the field of mathematics. Mandelbrot explained in clear terms the limitations of classical theory: Most of nature is very, very complicated.

The Perfect Bet: How Science and Math Are Taking the Luck Out of Gambling
by Adam Kucharski
Published 23 Feb 2016

The researchers found that as the number of players increased, this chaotic decision making became more common. When games are complicated, it seems that it may be impossible to anticipate players’ choices. Other patterns also emerged, including ones that had previously been spotted in real-life games. When mathematician Benoit Mandelbrot looked at financial markets in the early 1960s, he noticed that volatile periods in stock markets tended to cluster together. “Large changes tend to be followed by large changes,” he noted, “and small changes tend to be followed by small changes.” The appearance of “clustered volatility” has intrigued economists ever since.

The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
by Mario Livio
Published 23 Sep 2003

Investors know, however, that even with the application of all the bells and whistles of modern portfolio theory, which is supposed to maximize the returns for a decided-on level of risk, fortunes can be made or lost in a heartbeat. You may have noticed that Elliott's wave interpretation has as one of its ingredients the concept that each part of the 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 same when charts are enlarged or reduced to fit the same price and time scales.

Skin in the Game: Hidden Asymmetries in Daily Life
by Nassim Nicholas Taleb
Published 20 Feb 2018

Chapter 13 The Merchandising of Virtue Sontag is about Sontag—Virtue is what you do when nobody is looking—Have the guts to be unpopular—Meetings breed meetings—Call someone lonely on Saturdays after tennis Lycurgus, the Spartan lawmaker, responded to a suggestion to allow democracy there, saying “begin with your own family.” I will always remember my encounter with the writer and cultural icon Susan Sontag, largely because I met the great Benoit Mandelbrot on the same day. It took place in 2001, two months after the terrorist event of September, in a radio station in New York. Sontag, who was being interviewed, was piqued by the idea of a fellow who “studies randomness” and came to engage me. When she discovered that I was a trader, she blurted out that she was “against the market system” and turned her back to me as I was in mid-sentence, just to humiliate me (note here that courtesy is an application of the Silver Rule), while her assistant gave me a look as if I had been convicted of child killing.

New Dark Age: Technology and the End of the Future
by James Bridle
Published 18 Jun 2018

The reason, as he came to understand, was that the length of the border depended upon the tools used to measure it: as these became more accurate, the length actually increased, as smaller and smaller variations in the line were taken into account.41 Coastlines were even worse, leading to the realisation that it is in fact impossible to give a completely accurate account of the length of a nation’s borders. This ‘coastline paradox’ came to be known as the Richardson effect, and formed the basis for Benoît Mandelbrot’s work on fractals. It demonstrates, with radical clarity, the counterintuitive premise of the new dark age: the more obsessively we attempt to compute the world, the more unknowably complex it appears. 3 Climate There was a video on YouTube that I watched over and over again, until it got taken down.

Places of the Heart: The Psychogeography of Everyday Life
by Colin Ellard
Published 14 May 2015

In fact, the very name “fractal” is meant to convey this property of having a fractional dimensionality lying somewhere between whole numbers. Though this might seem a bit puzzling to picture, what it really means is that fractal objects defy some of the rules of conventional nonfractal geometry. In his original formulation of fractal dimension, the Polish mathematician Benoit Mandelbrot considered how one might go about measuring the length of a jagged coastline using a measuring stick. Because it contains a vast number of detailed curves and angles, the measured length of the coastline will depend on the length of the stick. As the stick becomes shorter and shorter, the length of the coastline will seem to become longer and longer.

Bad Data Handbook
by Q. Ethan McCallum
Published 14 Nov 2012

To see this idea in action, consider a classic technique for defining a complex graphical object by starting with two simple objects: “One begins with two shapes, an initiator and a generator…each stage of the construction begins with a broken line and consists in replacing each straight interval with a copy of the generator, reduced and displaced so as to have the same end points as those of the interval being replaced.” Benoît Mandelbrot[62] In just three iterations of this algorithm, we can create a famous shape known as the Koch snowflake.[63] Not so different than what just happened with our relational schema, is it? Our entities play the role of the “straight interval,” and the associative many-to-many entities act as the complexity generators.

The Loop: How Technology Is Creating a World Without Choices and How to Fight Back
by Jacob Ward
Published 25 Jan 2022

Goldman titled his article “Lindy’s Law,” and successive generations of writers seized on his theme as a way of predicting the longevity of certain ideas and works of art. This wasn’t just wannabe Late Night writers ruminating over bad coffee. Goldman’s concept launched a whole world of statistical thinking. The mathematician Benoit Mandelbrot updated the model in a 1982 book about fractals to posit that comedians who have made appearances in the past are more likely to make them in the future. Nicholas Taleb carried Mandelbrot’s concept into his book Black Swan, and in his book Antifragile put a specific math to the notion that as an idea survives, its longevity increases.

The Simulation Hypothesis
by Rizwan Virk
Published 31 Mar 2019

Figure 32: Fractal patterns resemble natural processes.78 According to the Fractal Foundation, “A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop.”79 Fractals have been around since the 1980s. Benoit Mandelbrot, as a young mathematician and researcher, found patterns of self-similarity at different scales in many different kinds of problems, ranging from error codes in telephone lines, to the pattern of prices of commodities in the markets, to the structure of a coastline. The coastline example is perhaps the best way to understand fractals.

Possible Minds: Twenty-Five Ways of Looking at AI
by John Brockman
Published 19 Feb 2019

The Fifth Generation: Artificial Intelligence and Japan’s Computer Challenge to the World was published in 1983. We had a code name for the project: “It’s coming, it’s coming!” But it didn’t come; it went. From that point on I’ve worked with researchers in nearly every variety of AI and complexity, including Rodney Brooks, Hans Moravec, John Archibald Wheeler, Benoit Mandelbrot, John Henry Holland, Danny Hillis, Freeman Dyson, Chris Langton, J. Doyne Farmer, Geoffrey West, Stuart Russell, and Judea Pearl. AN ONGOING DYNAMICAL EMERGENT SYSTEM From the initial meeting in Washington, Connecticut, to the present, I arranged a number of dinners and discussions in London and Cambridge, Massachusetts, as well as a public event at London’s City Hall.

Concentrated Investing
by Allen C. Benello
Published 7 Dec 2016

His research led him to fill three library shelves with books, including Adam Smith’s Wealth of Nations, John von Neumann and Oskar Morgenstern’s Theory of Games and Economic Behavior, Paul Samuelson’s Economics, and Fred Schwed’s Where Are the Customer’s Yachts? In a notebook Shannon recorded a varied list of thinkers, including French mathematician Louis Bachelier, Benjamin 74 Concentrated Investing Graham, and Benoit Mandelbrot. He took notes about margin trading; short selling; stop‐loss orders; the effects of market panics; capital gains taxes and transaction costs. The only surviving document from Shannon’s research is a mimeographed handout from one of the lectures he delivered at MIT in the spring term of 1956, in a class called Seminar of Information Theory.

Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets
by Nassim Nicholas Taleb
Published 1 Jan 2001

While it is clear that the world produces clusters it is also sad that these may be too difficult to predict (outside of physics) for us to take their models seriously. Once again the important fact is knowing the existence of these nonlinearities, not trying to model them. The value of the great Benoit Mandelbrot’s work lies more in telling us that there is a “wild” type of randomness of which we will never know much (owing to their unstable properties). Our Brain Our brain is not cut out for nonlinearities. People think that if, say, two variables are causally linked, then a steady input in one variable should always yield a result in the other one.

Descartes' Error: Emotion, Reason and the Human Brain
by António R. Damásio
Published 1 Jan 1994

If those symbols were not imageable, we would not know them and would not be able to manipulate them consciously. In this regard, it is interesting to observe that some insightful mathematicians and physicists describe their thinking as dominated by images. Often the images are visual, and they even can be somatosensory. Not surprisingly, Benoit Mandelbrot, whose life work is fractal geometry, says he always thinks in images.14 He relates that the physicist Richard Feynman was not fond of looking at an equation without looking at the illustration that went with it (and note that both equation and illustration were images, in fact). As for Albert Einstein, he had no doubts about the process: The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought.

Present Shock: When Everything Happens Now
by Douglas Rushkoff
Published 21 Mar 2013

They believe that, unlike traditional measurement and prediction, these nonlinear, systems approaches transcend the human inability to imagine the unthinkable. Even Black Swan author Nassim Taleb, who made a career of warning economists and investors against trying to see the future, believes in the power of fractals to predict the sudden shifts and wild outcomes of real markets. He dedicated the book to Benoit Mandelbrot. While fractal geometry can certainly help us find strong, repeating patterns within the market activity of the 1930s Depression, it did not predict the crash of 2007. Nor did the economists using fractals manage to protect their banks and brokerages from the systemic effects of bad mortgage packages, overleveraged European banks, or the impact of algorithmic trading on moment-to-moment volatility.

The Creativity Code: How AI Is Learning to Write, Paint and Think
by Marcus Du Sautoy
Published 7 Mar 2019

Perhaps the answer lies in the fact that they do not form a bridge between two conscious worlds. Computer-generated fractals have nonetheless made their creators big money, as fractals have proven to be a highly effective way to simulate the natural world. In his seminal book The Fractal Geometry of Nature, Benoit Mandelbrot explained how Nature uses fractal algorithms to make ferns, clouds, waves, mountains. It was reading this book that inspired Loren Carpenter, an engineer working at Boeing, to experiment with code to simulate natural worlds on the computer. Using the Boeing computers at night-time, he put together a two-minute animation of a fly-through of his computer-generated fractal landscape.

The Cultural Logic of Computation
by David Golumbia
Published 31 Mar 2009

The CCS should be remembered as the first major center to use the words cognitive science, and once again it is fascinating to note who else backed this project: Over the years [the CCS] became a gathering place for those scientists who were most active in the blending of psychology, linguistics, computer modeling, philosophy, and information theory that Miller and Bruner were backing. Noam Chomsky, Nelson Goodman, Benoit Mandelbrot, Donald Norman, Jerrold Katz, Thomas Bever, Eric Lennenberg, and Joseph Weizenbaum were only a few of the dozens of visitors who spent a year or more at the Center between 1960 and 1966. More than $2 million in grant money flowed into the Center’s coffers, mostly from the Carnegie Corporation, the National Institutes of Health, and ARPA.

Growth: From Microorganisms to Megacities
by Vaclav Smil
Published 23 Sep 2019

Jaromír Korčák called attention to the duality of statistical distribution, with the outcome of organic growth organized in normal fashion, while the distribution of the planet’s physical characteristics—area and depth of lakes, size of islands, area of watersheds, length of rivers—follows inverse power law with distributions highly skewed leftward (Korčák 1938 and 1941). Korčák’s law was later made better known, via Fréchet (1941), by Benoit Mandelbrot in his pioneering work on fractals (Mandelbrot 1967, 1975, 1977, 1982). But a recent reexamination of Korčák’s law concluded that his ranked properties cannot be described with a single power-law exponent and hence the law is not strictly valid even for sets consisting of strictly similar fractal objects presented in his original publications (Imre and Novotný 2016).

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In the equation N = 10a−bM a indicates the activity rate (how many earthquakes of a given magnitude in a year) and b is usually close to 1 for interplate events but it is higher along oceanic ridges and lower for intraplate earthquakes. Quincy Wright (1942) and Lewis F. Richardson (1948) used power law to explain the variation of the frequency of fatal conflicts with their magnitude. And Benoit Mandelbrot’s pioneering studies of self-similarity and fractal structures further expanded the applications of power laws: after all, the “probability distribution of a self-similar random variable X must be of the form Pr(X>x) = x-D, which is commonly called hyperbolic or Pareto distribution” (Mandelbrot 1977, 320).

The Road to Ruin: The Global Elites' Secret Plan for the Next Financial Crisis
by James Rickards
Published 15 Nov 2016

Good Money Part I: The New World. Indianapolis: Liberty Fund, 1999. ———. Good Money Part II: The Standard. Indianapolis: Liberty Fund, 1999. Hudson, Michael. Killing the Host: How Financial Parasites and Debt Destroy the Global Economy. Bergenfield, NJ: ISLET, 2015. Hudson, Richard L., and Benoit Mandelbrot. The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books, 2004. Hui, Pak Ming, Paul Jefferies, and Neil F. Johnson. Financial Market Complexity: What Physics Can Tell Us About Market Behavior. Oxford: Oxford University Press, 2003. Jensen, Henrik Jeldtoft.

Irrational Exuberance: With a New Preface by the Author
by Robert J. Shiller
Published 15 Feb 2000

The literature on applications of chaos theory to economics usually does not stress the kind of price feedback model discussed here, but it may nonetheless offer some insights into the sources of complexity in financial markets. See Michael Boldrin and Michael Woodford, “Equilibrium Models Displaying Endogenous Fluctuations and Chaos: A Survey,” Journal of Monetary Economics, 25(2) (1990): 189–222, for a survey of this literature. See also Benoit Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk (New York: Springer-Verlag, 1997); and Brian Arthur, John H. Holland, Blake LeBaron, Richard Palmer, and Paul Tayler, “Asset Pricing under Endogenous Expectations in an Artificial Stock Market,” in W. B. Arthur, S. Durlauf, and D.

Is the Internet Changing the Way You Think?: The Net's Impact on Our Minds and Future
by John Brockman
Published 18 Jan 2011

This year, I enlisted the aid of Hans Ulrich Obrist, curator of the Serpentine Gallery in London, and the artist April Gornik, one of the early members of the Reality Club, to help broaden the Edge conversation—or, rather, to bring it back to where it was in the late 1980s and early 1990s, when April gave a talk at a Reality Club meeting and discussed the influence of chaos theory on her work, and Benoit Mandelbrot showed up to discuss fractal theory. Every artist in New York City wanted to be there. What then happened was very interesting. When the Reality Club went online as Edge, the scientists were all on e-mail—and the artists weren’t. Thus did Edge, surprisingly, become a science site, whereas my own background (beginning in 1965, when Jonas Mekas hired me to manage the Film-Makers’ Cinematheque) was in the visual and performance arts.

Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street
by William Poundstone
Published 18 Sep 2006

Part of the notebook is devoted to the roulette device and part to a wildly disconnected set of stock market musings. Shannon wondered about the statistical structure of the market’s random walk and whether information theory could provide useful insights. He mentions such diverse names as Bachelier, (Benjamin) Graham and (David) Dodd, (John) Magee, A. W. Jones, (Oskar) Morgenstern, and (Benoit) Mandelbrot. He considered margin trading and short-selling; stop-loss orders and the effects of market panics; capital gains taxes and transaction costs. Shannon graphs short interest in Litton Industries (shorted shares vs. price: the values jump all over with no evident pattern). He notes such success stories as Bernard Baruch, the Lone Wolf, who ran about $10,000 into a million in about ten years, and Hetty Green, the Witch of Wall Street, who ran a million into a hundred million in thirty years.

A Place of My Own: The Architecture of Daydreams
by Michael Pollan
Published 15 Jan 1997

On Weathering (Cambridge: MIT Press, 1993). My principal sources on trees and woods were Herbert L. Edlin’s What Wood Is That? A Manual of Wood Identification (New York: Viking Press, 1969) and Donald Culross Peattie’s A Natural History of Trees of Eastern and Central North America (Boston: Houghton Mifflin, 1966). Benoit Mandelbrot’s ideas about architectural ornament are discussed briefly in James Gleick’s Chaos: Making a New Science (New York: Viking, 1987). On the history of the study and the rise of the modern individual the key work is A History of Private Life, edited by Philippe Ariès and Georges Duby. See Volume III, Passions of the Renaissance, especially Ariès’s introduction, as well as “The Refuges of Intimacy” by Orest Ranum, and “The Practical Impact of Writing” by Roger Chartier.

Trillions: How a Band of Wall Street Renegades Invented the Index Fund and Changed Finance Forever
by Robin Wigglesworth
Published 11 Oct 2021

And that’s how the precocious former jock ended up at the University of Chicago. Aside from a two-year stint as a visiting professor at the University of Leuven in Belgium in the mid-1970s, Fama has stayed true to Chicago ever since the fateful call, and is still teaching in his eighties. It was at Chicago that Fama first met Benoit Mandelbrot, a brilliant Polish-French American mathematician. The peripatetic polymath would occasionally visit the university and give presentations to its graduate students, and ended up taking many long walks around the university’s quadrangles with the young Fama. Crucially, it was Mandelbrot who told the young Italian American about the apparent randomness of financial markets, and Bachelier’s groundbreaking work over half a century earlier.

The Man Who Solved the Market: How Jim Simons Launched the Quant Revolution
by Gregory Zuckerman
Published 5 Nov 2019

A quarter century later, legendary New York Times financial columnist Floyd Norris called it, “the beginning of the destruction of markets by dumb computers. Or, to be fair to the computers, by computers programmed by fallible people and trusted by people who did not understand the computer programs’ limitations. As computers came in, human judgment went out.” During the 1980s, Professor Benoit Mandelbrot—who had demonstrated that certain jagged mathematical shapes called fractals mimic irregularities found in nature—argued that financial markets also have fractal patterns. This theory suggested that markets will deliver more unexpected events than widely assumed, another reason to doubt the elaborate models produced by high-powered computers.

The Master Algorithm: How the Quest for the Ultimate Learning Machine Will Remake Our World
by Pedro Domingos
Published 21 Sep 2015

Gary Marcus, Adam Marblestone, and Tom Dean make the case against in “The atoms of neural computation” (Science, 2014). “The unreasonable effectiveness of data,” by Alon Halevy, Peter Norvig, and Fernando Pereira (IEEE Intelligent Systems, 2009), argues for machine learning as the new discovery paradigm. Benoît Mandelbrot explores the fractal geometry of nature in the eponymous book* (Freeman, 1982). James Gleick’s Chaos (Viking, 1987) discusses and depicts the Mandelbrot set. The Langlands program, a research effort that seeks to unify different subfields of mathematics, is described in Love and Math, by Edward Frenkel (Basic Books, 2014).

Wireless
by Charles Stross
Published 7 Jul 2009

(True names have power, so the Laundry is big on call by reference, not call by value; I’m no more “Bob Howard” than the “Alan Turing” in room two is the father of computer science and applied computational demonology.) She continues. “The real Alan Turing would be nearly a hundred by now. All our long-term residents are named for famous mathematicians. We’ve got Alan Turing, Kurt Godel, Georg Cantor, and Benoit Mandelbrot. Turing’s the oldest, Benny is the most recent—he actually has a payroll number, sixteen.” I’m in five digits—I don’t know whether to laugh or cry. “Who’s the nameless one?” I ask. “That would be Georg Cantor,” she says slowly. “He’s probably in room four.” I bend over the indicated periscope, remove the brass cap, and peer into the alien world of the nameless K.

The New Trading for a Living: Psychology, Discipline, Trading Tools and Systems, Risk Control, Trade Management
by Alexander Elder
Published 28 Sep 2014

Channels show where to expect support and resistance in the future. Channels help identify buying and selling opportunities and avoid bad trades. The original research into trading channels was conducted by J. M. Hurst and described in his 1970 book, The Profit Magic of Stock Transaction Timing. The late great mathematician Benoit Mandelbrot was hired by the Egyptian government to create a mathematical model of cotton prices—the main agricultural export of that country. After extensive study, the scientist made this finding: “prices oscillate above and below value.” It may sound simple, but in fact it's profound. If we accept this mathematical finding and if we have the means to define value and measure an average oscillation, we'll have a trading system.

Human Frontiers: The Future of Big Ideas in an Age of Small Thinking
by Michael Bhaskar
Published 2 Nov 2021

It encompassed mathematics, meteorology, population ecology, epidemiology, physics and economics, underwritten by advances in computing and computer science and applied to topics as various as the organs of the human body, atmospheric storms and a beam of particles. Names like Edward Lorenz, Mary Cartright, Benoit Mandelbrot and Mitchell Feigenbaum were all closely connected, but it didn't have one great progenitor in the manner of a Darwin or Pasteur. The study of complexity is itself, perhaps unsurprisingly, complex, transdisciplinary, composed of different foci and sub-ideas, while nonetheless adding up to something big and revolutionary.

Red-Blooded Risk: The Secret History of Wall Street
by Aaron Brown and Eric Kim
Published 10 Oct 2011

The view of quantitative finance described in Red-Blooded Risk has a lot of overlap with two pathbreaking but eccentric works: The Handbook of Portfolio Mathematics: Formulas for Optimal Allocation & Leverage by Ralph Vince and Finding Alpha: The Search for Alpha When Risk and Return Break Down by Eric Falkenstein. A more famous pathbreaking and eccentric work is Benoit Mandelbrot’s The (Mis)behavior of Markets. Two of the best books on the future of finance are The New Financial Order: Risk in the 21st Century by Robert J. Shiller and Financing the Future: Market-Based Innovations for Growth by Franklin Allen and Glenn Yago. Both cover quite a bit of history to ground their predictions in something solid.

Whole Earth: The Many Lives of Stewart Brand
by John Markoff
Published 22 Mar 2022

The event, held in a hotel conference center in Monterey in February of 1984, was a little-noticed affair attended by a relatively intimate group (compared to later conferences) of 250 artists, writers, musicians, corporate execs, and scientists united by their “faith in the computer.”[18] IBM mathematician Benoit Mandelbrot and Megatrends futurist John Naisbitt both spoke, but it was Negroponte who stole the show. Dressed in a dapper gray suit and tie and with a rich mane of longish hair, he showed off in his “TED Talk” (the term had yet to become a marque—and, in some quarters, a pigeonhole) a variety of futuristic technologies for interacting with computers, including touch screen manipulation, which would not become commonplace until a quarter century later with the introduction of the iPhone.

Ways of Being: Beyond Human Intelligence
by James Bridle
Published 6 Apr 2022

On each measurement, the reading would get more accurate, with more and more of the coastline accounted for. But the result, as Richardson realized, was not that the measurement converged on the correct answer but rather, the more closely it was measured, the longer it got. What Richardson had discovered was what the mathematician Benoit Mandelbrot would later term ‘fractals’: structures which repeat to infinite complexity. Instead of resolving into order and clarity, ever-closer examination reveals only more, and more splendid, detail and variation.26 The Richardson effect applies to biology, archaeology, evolution and, it seems, to life itself.

Visions of Inequality: From the French Revolution to the End of the Cold War
by Branko Milanovic
Published 9 Oct 2023

Philippe Aghion and Patrick Bolton, “Distribution and Growth in Models of Imperfect Capital Markets,” European Economic Review 36 (1992): 603–621. 58 . Examples of studies of stochastic models of income distribution include D. G. Champernowne, The Distribution of Income between Persons (Cambridge: Cambridge University Press, 1973); Benoit Mandelbrot, “The Pareto- L é vy Law and the Distribution of Income,” International Economic Review 1, no. 2 (1960), 79–106; Thomas Mayer, “The Distribution of Ability and Earnings,” Review of Economics and Statistics 42, no. 2 (1960): 189–195. 59 . See, for example, Christopher Bliss, Capital Theory and the Distribution of Income (Amsterdam: North-Holland, 1975); Piero Garegnani, “Heterogeneous Capital, the Production Function and the Theory of Distribution,” Review of Economic Studies 37, no. 3 (1970): 407–436. 60 .

Power, Sex, Suicide: Mitochondria and the Meaning of Life
by Nick Lane
Published 14 Oct 2005

Their densely mathematical model was published in Science in 1997, and the ramifications (if not the maths) swiftly captured the imagination of many. The fractal tree of life Fractals (from the Latin fractus, broken) are geometric shapes that look similar at any scale. If a fractal is broken into its constituent parts, each part still looks more or less the same, because, as the pioneer of fractal geometry Benoit Mandelbrot put it, ‘the shapes are made of parts similar to the whole in some way’. Fractals can be formed randomly by natural forces such as wind, rain, ice, erosion, and gravity, to generate natural fractals, like mountains, clouds, rivers, and coastlines. Indeed, Mandelbrot described fractals as ‘the geometry of nature’, and in his landmark paper, published in Science, in 1967, he applied this approach to the question advanced in its title: How Long is the Coast of Britain?

Life's Greatest Secret: The Race to Crack the Genetic Code
by Matthew Cobb
Published 6 Jul 2015

For a biologist, argued Lwoff, the only meaning of information was ‘a sequence of small molecules and the set of functions they carry out’.30 Wiener and the philosophers who were present could not see what the problem was, thereby inadvertently illustrating the gulf between the information theoreticians and the biologists. Similar mutual incomprehension was revealed in the other sessions, which were often fractious. The mathematician Benoît Mandelbrot suggested that such information-focused cross-disciplinary meetings were pointless: The implications of the strict meaning of information have sufficiently explored for its consequences to be quite clear. What remains is so difficult that it can usefully be discussed only in private … we must consider that its scientific usefulness has ceased, at least for the time being’.31 Alongside these rather sterile plenary discussions there were workshops in which experts in the various fields explored their topic in more detail.

In Pursuit of the Perfect Portfolio: The Stories, Voices, and Key Insights of the Pioneers Who Shaped the Way We Invest
by Andrew W. Lo and Stephen R. Foerster
Published 16 Aug 2021

He later mused, “I wonder what path my professional life would have taken if Jeff didn’t answer the phone that day. Serendipity!”12 Fama attended Chicago’s PhD program in economics between 1960 and 1964. In his second year, near the completion of his coursework, he began to attend the department’s Econometrics Workshop. An occasional presenter was Benoit Mandelbrot, a highly regarded mathematician on staff as a researcher at the IBM Thomas J. Watson Research Center and a visiting professor at Harvard University, today best known for his work on fractals and their irregular geometry. Fama enjoyed strolling the campus with Mandelbrot and learned much about probability distributions from him, including Mandelbrot’s research on cotton prices.

The Volatility Smile
by Emanuel Derman,Michael B.Miller
Published 6 Sep 2016

We can use Equation 24.48 to get the approximate formula for the smile in Figure 24.6, which is (√ √ 𝜆 𝜏e−𝜆𝜏 J 𝜋 ) ( ) K 1 + √ ln 𝛴 ≈𝜎+ 2 𝜎 𝜏 S S ( ) K ≈ 0.102 + 0.56 × ln S (24.50) which is a good approximation to the exact results near at-the-money. 416 THE VOLATILITY SMILE FURTHER THOUGHTS AND READING Merton’s model of jump-diffusion regards jumps as “abnormal” market events that have to be superimposed upon “normal” diffusion. The view that the market has two regimes of behavior, normal and abnormal, is regarded as contrived both by Benoit Mandelbrot and by Eugene Stanley and his econophysics collaborators. To paraphrase their view, a single model rather than a mixture of “normal” and “abnormal” models should ideally explain all events. END-OF-CHAPTER PROBLEMS 24-1. Estimate the price of a 24,000 strike put with two weeks to expiration on the Hang Seng Index (HSI).

A Splendid Exchange: How Trade Shaped the World
by William J. Bernstein
Published 5 May 2009

Goitein, A Mediterranean Society (Berkeley: University of California Press, 1967), I: 347-348. 7. Ibid., 298. 8. Ibid., 299-300. 9. Ibid., 340-342. 10. Ibid., 219. Only in the twentieth century did economists begin to fully appreciate the unpredictability of market prices. By a strange coincidence, the founder of chaos theory, Benoit Mandelbrot, drew his original inspiration by connecting the pattern of cotton prices with that of the flooding pattern of the Nile. 11. The dinar, like most of the standard gold coins of the premodern period, weighed about one-eighth of an ounce, worth about eighty dollars at current value. Thus, an annual income of one hundred dinars corresponds to about $8,000 per year in today's currency. 12.

Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures
by Frank J. Fabozzi
Published 25 Feb 2008

The immense threat radiating from heavy tails in stock return distributions made industry professionals aware of the urgency to take them serious and reflect them in their models. Many distributional alternatives providing more realistic chances to severe price movements have been presented earlier, such as the Student’s t in Chapter 11 or GEV distributions earlier in this chapter, for example. In the early 1960s, Benoit Mandelbrot suggested as a distribution for commodity price changes the class of stable distributions. The reason is that, through their particular parameterization, they are capable of modeling moderate scenarios as supported by the normal distribution as well as extreme ones beyond the scope of most of the distributions that we have presented in this chapter.

New Market Wizards: Conversations With America's Top Traders
by Jack D. Schwager
Published 28 Jan 1994

Why do you feel such techniques are more appropriate for trading system analysis? Because I believe that price distributions are pathological. In what way? As one example, price distributions have more variance [a statistical measure of the variability in the data] than one would expect on the basis of normal distribution theory. Benoit Mandelbrot, the originator of the concept of fractional dimension, has conjectured that price change distributions actually have infinite variance. The sample variance [i.e., the implied variability in prices] just gets larger and larger as you add more data. If this is true, then most standard statistical techniques are invalid for price data applications.

The Economists' Hour: How the False Prophets of Free Markets Fractured Our Society
by Binyamin Appelbaum
Published 4 Sep 2019

Fama, “Efficient Capital Markets: A Review of Theory and Empirical Work,” Journal of Finance 25, no. 2 (1970). 15. The theory said nobody, no matter how smart, could predict the future movements of stock prices based on the information already in existence. The movement would be determined by what happened next. The economist Benoit Mandelbrot compared markets to a drunken man in an open field: he might stumble in any direction; he might double back on his own tracks. The only useful information about where he would end up was where he stood at the start. The theory actually comes in three progressively stronger formulations. The weakest version says past price movements cannot be used to forecast future price movements.

More Money Than God: Hedge Funds and the Making of a New Elite
by Sebastian Mallaby
Published 9 Jun 2010

The efficient-market hypothesis had always been based on a precarious assumption: that price changes conformed to a “normal” probability distribution—the one represented by the familiar bell curve, in which numbers at and near the median crop up frequently while numbers in the tails of the distribution are rare to the point of vanishing. Even in the early 1960s, a maverick mathematician named Benoit Mandelbrot argued that the tails of the distribution might be fatter than the normal bell curve assumed; and Eugene Fama, the father of efficient-market theory, who got to know Mandelbrot at the time, conducted tests on stock-price changes that confirmed Mandelbrot’s assertion. If price changes had been normally distributed, jumps greater than five standard deviations should have shown up in daily price data about once every seven thousand years.

I Am a Strange Loop
by Douglas R. Hofstadter
Published 21 Feb 2011

My explorations did not teach me that any shape whatsoever can arise as a result of video feedback, but they did show me that I had entered a far richer universe of possibilities than I had expected. Today, this visual richness reminds me of the amazing visual universe discovered around 1980 by mathematician Benoit Mandelbrot when he studied the properties of the simple iteration defined by z → z2 + c, where c is a fixed complex number and z is a variable complex number whose initial value is 0. This is a mathematical feedback loop where one value of z goes in and a new value comes out, ready to be fed back in again, just as in audio or video feedback.

Clojure Programming
by Chas Emerick , Brian Carper and Christophe Grand
Published 15 Aug 2011

* * * [349] See http://en.wikipedia.org/wiki/Mandlebrot_set for a gentle introduction to the Mandelbrot Set, the mathematics behind it, and how you can go about generating visualizations of it. We’d also be remiss if we didn’t point you toward Jonathan Coulton’s fantastic song and music video about the Mandelbrot Set and its creator/discoverer Benoît Mandelbrot: http://www.youtube.com/watch?v=ES-yKOYaXq0. [350] Simpler implementations are possible; for example, by using the iterate function to lazily calculate the result of the complex polynomial, and take-ing only as many results from the head of that lazy seq as dictated by our maximum iteration count.

The Signal and the Noise: Why So Many Predictions Fail-But Some Don't
by Nate Silver
Published 31 Aug 2012

Watson Center—a beautiful, crescent-shaped, retro-modern building overlooking the Westchester County foothills. In its lobby are replicas of early computers, like the ones designed by Charles Babbage. While the building shows a few signs of rust—too much wood paneling and too many interior offices—many great scientists have called it home, including the mathematician Benoit Mandelbrot, and Nobel Prize winners in economics and physics. I visited the Watson Center in the spring of 2010 to see Murray Campbell, a mild-mannered and still boyish-looking Canadian who was one of the chief engineers on the project since its days as Deep Thought at Carnegie Mellon. (Today, Campbell oversees IBM’s statistical modeling department.)

Dead or Alive
by Tom Clancy and Grant (CON) Blackwood
Published 7 Dec 2010

His government salary had long since topped him out as a Senior Executive Service genius, and indeed he still collected his reasonably generous government pension. But he loved the action and had snapped up the offer to join The Campus within seconds of its being made. He was, professionally, a mathematician, with a doctorate from Harvard, where he’d studied under Benoit Mandelbrot himself, and he occasionally lectured at MIT and Caltech as well in his area of expertise. Biery was a geek through and through, right down to the heavy black-rimmed glasses and doughy complexion, but he kept The Campus’s electronic gears oiled and the machines purring. “Compartmentalization?”

The Gene: An Intimate History
by Siddhartha Mukherjee
Published 16 May 2016

“Important Biological Objects Come in Pairs” One could not be a successful scientist without realizing that, in contrast to the popular conception supported by newspapers and the mothers of scientists, a goodly number of scientists are not only narrow-minded and dull, but also just stupid. —James Watson It is the molecule that has the glamour, not the scientists. —Francis Crick Science [would be] ruined if—like sports—it were to put competition above everything else. —Benoit Mandelbrot Oswald Avery’s experiment achieved another “transformation.” DNA, once the underdog of all biological molecules, was thrust into the limelight. Although some scientists initially resisted the idea that genes were made of DNA, Avery’s evidence was hard to shrug off (despite three nominations, however, Avery was still denied the Nobel Prize because Einar Hammarsten, the influential Swedish chemist, refused to believe that DNA could carry genetic information).

Debunking Economics - Revised, Expanded and Integrated Edition: The Naked Emperor Dethroned?
by Steve Keen
Published 21 Sep 2011

Complexity theorists argue that the economy demonstrates similar attributes, and these are what give rise to the cycles which are a self-evident aspect of real-world economies. Econophysics substantially adds to the contribution made by the early proponents of complexity in economics – such as Richard Goodwin (Goodwin 1990, 1991), Benoit Mandelbrot (Mandelbrot 1971, 2005), Hans-Walter Lorenz (Lorenz 1987a, 1987b, 1989), Paul Ormerod (Ormerod 1997, 2001, 2004); Ormerod and Heineike (2009), Carl Chiarella (Chiarella and Flaschel 2000, Chiarella, Dieci et al. 2002, Chiarella et al. 2003) and myself, among many others – by bringing both the techniques and the empirical mindset of physicists to bear upon economic data.

Accessory to War: The Unspoken Alliance Between Astrophysics and the Military
by Neil Degrasse Tyson and Avis Lang
Published 10 Sep 2018

Today (presumably taking into account only some of the ins and outs), according to the Ordnance Survey, Britain’s national mapping agency, “The coastline length around mainland Great Britain is 11,072.76 miles,” www.ordnancesurvey.co.uk/oswebsite/freefun/didyouknow/ (accessed May 17, 2010). But as Benoit Mandelbrot famously proposed in “How Long Is the Coastline of Britain?” there is no end of different numbers. Nevertheless, Pytheas was certainly on the right track compared with contemporaries of his, such as the disbelieving Strabo. 20.Roseman, Pytheas, 7–20, writes that eighteen known ancient writers referred to Pytheas by name between 300 BC and AD 550, notably Eratosthenes, Hipparkhos, Polybius, Strabo, and Pliny the Elder.

The Singularity Is Near: When Humans Transcend Biology
by Ray Kurzweil
Published 14 Jul 2005

To understand this, let's first consider the fractal nature of the brain's organization, which I discussed in chapter 2. A fractal is a rule that is iteratively applied to create a pattern or design. The rule is often quite simple, but because of the iteration the resulting design can be remarkably complex. A famous example of this is the Mandelbrot set devised by mathematician Benoit Mandelbrot.20 Visual images of the Mandelbrot set are remarkably complex, with endlessly complicated designs within designs. As we look at finer and finer detail in an image of the Mandelbrot set, the complexity never goes away, and we continue to see ever finer complication. Yet the formula underlying all of this complexity is amazingly simple: the Mandelbrot set is characterized by a single formula Z = Z2 + C, in which Z is a "complex" (meaning two-dimensional) number and C is a constant.

The Man Who Knew: The Life and Times of Alan Greenspan
by Sebastian Mallaby
Published 10 Oct 2016

At that time, the fathers of modern portfolio theory confronted a highly inconvenient truth: contrary to their efficient-market assumptions, price changes in asset markets do not follow the “normal distribution” depicted by a bell curve; rather, very large price moves occur far more frequently than the thin tails of the bell curve anticipate. At first the efficient marketers responded open-mindedly to this objection, acknowledging that its main proponent, the maverick mathematician Benoit Mandelbrot, was right. But then they swept Mandelbrot’s protests under the carpet because his message was too difficult to live with. Deprived of their bell-curve assumption, the efficient marketers’ mathematical techniques would cease to work. “Mandelbrot, like Prime Minister Churchill before him, promises us not utopia but blood, sweat, toil and tears,” Paul Cootner, an efficient marketer, objected.

Surfaces and Essences
by Douglas Hofstadter and Emmanuel Sander
Published 10 Sep 2012

Let’s not forget that between the integers there are plenty of other numbers (for example, 1/2 and 5/17 and 3.14159265358979…, etc.), and mathematicians in the early twentieth century who were interested in abstract spaces — especially the German mathematician Felix Haussdorff — came up with ways to generalize the concept of dimensionality, thus leading to the idea of spaces having, say, 0.73 dimensions or even π dimensions. These discoveries later turned out to be ideally suited for characterizing the dimensionality of “fractal objects”, as they were dubbed by the Franco–Polish mathematician Benoît Mandelbrot. After such richness, one might easily presume that there must be spaces having a negative or imaginary number of dimensions — but oddly enough, despite the appeal of the idea, this notion has not yet been explored, or at any rate, if it has, we are ignorant of the fact. But the mindset of today’s mathematicians is so generalization-prone that even the hint of such an idea might just launch an eager quest for all the beautiful new abstract worlds that are implicit in the terms.

The Sum of All Fears
by Tom Clancy
Published 2 Jan 1989

It has a long and distinguished history that has benefited from another traditional Russian strength, a fascination with theoretical mathematics. The relationship between ciphers and mathematics is a logical one, and the most recent manifestation of this was the work of a bearded, thirtyish gnome of a man who was fascinated with the work of Benoit Mandelbrot at Harvard University, the man who had effectively invented fractal geometry. Uniting this work with that of MacKenzie's work on Chaos Theory at Cambridge University in England, the young Russian genius had invented a genuinely new theoretical way of looking at mathematical formulae. It was generally conceded by that handful of people who understood what he was talking about that his work was easily worth a Planck Medal.

The Master and His Emissary: The Divided Brain and the Making of the Western World
by Iain McGilchrist
Published 8 Oct 2012

L’Orange, 1965, p. 3. 151. ibid., pp. 3–8. 152. ibid., pp. 9–11. 153. Fractality is the property of forms as diverse as plants, river systems, coast lines, snowflakes and blood vessels that dictates that their form at higher levels of magnification replicates their form at lower levels. Although the term is modern, and derives from the mathematics of Benoît Mandelbrot in the mid-1970s, Leibniz may already have intuited, possibly on the basis of microscope findings, that nature is fractal: see Leibniz, 1992, §67–8, pp. 25–6, and commentary on pp. 41 & 234 ff. Elsewhere in this aphoristic late work, Leibniz relates his description of these worlds within worlds that formed part of his monadology to two further concepts of relevance for the theme of this book: the way that each body mirrors its environing universe, and each soul mirrors its environing body (and consequently the entire universe) (§61–2); and the way in which ‘all bodies are in a perpetual flux, like rivers, and some parts enter into them and some pass out continually’ (§71–2). 154.

Corporate Finance: Theory and Practice
by Pierre Vernimmen , Pascal Quiry , Maurizio Dallocchio , Yann le Fur and Antonio Salvi
Published 16 Oct 2017

A number of “anomalies” that tend to go against the efficiency of markets have been highlighted: Excess volatility. The first issue with efficient market theory seems very intuitive: how can markets be so volatile? Information on Sanofi is not published every second. Nevertheless, the share price does move at each instant. There seems to be some kind of noise around fundamental value. As described by Benoit Mandelbrot, who first used fractals in economics, prices evolve in a discrete way rather than in a continuous manner. Dual listing and closed-end funds. Dual listings are shares of twin companies listed on two different markets. Their stream of dividends is, by definition, identical but we can observe that their price can differ over a long period of time.