mandelbrot fractal

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pages: 364 words: 101,286

The Misbehavior of Markets by Benoit Mandelbrot


Albert Einstein, asset allocation, Augustin-Louis Cauchy, Benoit Mandelbrot, Big bang: deregulation of the City of London, Black-Scholes formula, British Empire, Brownian motion, buy low sell high, capital asset pricing model, carbon-based life, discounted cash flows, diversification, double helix, Edward Lorenz: Chaos theory, Elliott wave, equity premium, Eugene Fama: efficient market hypothesis, Fellow of the Royal Society, full employment, Georg Cantor, Henri Poincaré, implied volatility, index fund, informal economy, invisible hand, John Meriwether, John von Neumann, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, market bubble, market microstructure, Myron Scholes, new economy, paper trading, passive investing, Paul Lévy, Paul Samuelson, Plutocrats, plutocrats, price mechanism, quantitative trading / quantitative finance, Ralph Nelson Elliott, RAND corporation, random walk, risk tolerance, Robert Shiller, Robert Shiller, short selling, statistical arbitrage, statistical model, Steve Ballmer, stochastic volatility, transfer pricing, value at risk, Vilfredo Pareto, volatility smile

Comptes Rendus (Paris) : 278A ; 289-292 et 355-358. • Reprint : Chapter N16 of Mandelbrot 1999a. Mandelbrot, Benoit B. 1975. Les objets fractals : forme, hasard et dimension. Paris : Flammarion. Mandelbrot, Benoit B. 1982. The Fractal Geometry of Nature. New York: W.H. Freeman & Co. Mandelbrot, Benoit B. 1985. Self-affine fractals and fractal dimension. Physica Scripta 32: 257-260. • Reprint: Dynamics of Fractal Surfaces. Edited by Fereydoon Family & Tamas Vicsek. Singapore: World Scientific, 1991, 11-20. • Reprint Chapter H21 of Mandelbrot 2002. Mandelbrot, Benoit B. 1986. Self-affine fractal sets, I: The basic fractal dimensions, II: Length and area measurements, III: Hausdorff dimension anomalies and their implications. Fractals in Physics. Edited by Luciano Pietronero & Erio Tosatti, Amsterdam: North-Holland, 3-28. • Reprint of Part I: Dynamics of Fractal Surfaces.

. • Reprint of Part I: Dynamics of Fractal Surfaces. Edited by Fereydoon Family & Tamas Vicsek. Singapore: World Scientific, 1991, 21-36. • Reprint in Chapters H22, H23, H24 of Mandelbrot 2002. Mandelbrot, Benoit B. 1990. Limit lognormal multifractal measures. Frontiers of Physics: Landau Memorial Conference (Tel Aviv, 1988). Edited by E. A. Gotsman et al. New York: Pergamon, 309-340. Mandelbrot, Benoit B. 1997a. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. New York: Springer-Verlag. Mandelbrot, Benoit B. 1997b. Fractales, hasard et finance. Paris: Flammarion. Mandelbrot, Benoit B. 1997c. Three fractal models in finance: Discontinuity, concentration, risk. Economic Notes (Banca Monte dei Paschi di Siena SpA) 26 (2): 197-212. Mandelbrot, Benoit B. 1997d. Les fractales et la bourse. Pour la Science 242: 16-17.

. • Related paper: Journal of Statistical Physics 110, 2003, 739-777. Mandelbrot, Benoit B. 2004a. Fractals and Chaos: The Mandelbrot Set and Beyond. New York: Springer Verlag. Mandelbrot, Benoit B. 2004b. Updated reprint of Mandelbrot 1997a. Mandelbrot, Benoit B., Adlai Fisher, and Laurent Calvet. 1997. A multifractal model of asset returns. Cowles Foundation Discussion Paper 1164. • See also under Calvet and Fisher. Mandelbrot, Benoit B. and H.M. Taylor. 1967. On the distribution of stock price differences. Operations Research 15: 1057-1062. • Reprint: Chapter E21 of Mandelbrot 1997a. Mandelbrot, Benoit B. and J.W. Van Ness. 1968. Fractional Brownian motions, fractional noises and applications. SIAM Review 10: 422-437. • Reprint: Chapter H11 of Mandelbrot 2002. Mandelbrot, Benoit B. and James R. Wallis 1968. Noah, Joseph and operational hydrology.

pages: 396 words: 112,748

Chaos by James Gleick


Benoit Mandelbrot, butterfly effect, cellular automata, Claude Shannon: information theory, discrete time, Edward Lorenz: Chaos theory, experimental subject, Georg Cantor, Henri Poincaré, Isaac Newton, iterative process, John von Neumann, Louis Pasteur, mandelbrot fractal, Murray Gell-Mann, Norbert Wiener, pattern recognition, Richard Feynman, Richard Feynman, Stephen Hawking, stochastic process, trade route

“YOU OBTAIN AN INCREDIBLE VARIETY “Julia Sets and the Mandelbrot Set,” p. 161. IN 1979 MANDELBROT DISCOVERED Mandelbrot, Laff, Hubbard. A first-person account by Mandelbrot is “Fractals and the Rebirth of Iteration Theory,” in The Beauty of Fractals, pp. 151–60. AS HE TRIED CALCULATING Mandelbrot; The Beauty of Fractals. MANDELBROT STARTED WORRYING Mandelbrot. NO TWO PIECES ARE “TOGETHER” Hubbard. “EVERYTHING WAS VERY GEOMETRIC” Peitgen. AT CORNELL, MEANWHILE Hubbard. RICHTER HAD COME TO COMPLEX SYSTEMS Richter. “IN A BRAND NEW AREA” Peitgen. “RIGOR IS THE STRENGTH” Peitgen. FRACTAL BASIN BOUNDARIES Yorke; a good introduction, for the technically inclined, is Steven W. MacDonald, Celso Grebogi, Edward Ott, and James A. Yorke, “Fractal Basin Boundaries,” Physica 17D (1985), pp. 125–83.

Review articles on applications of fractals have grown too common to list, but two useful examples are Leonard M. Sander, “Fractal Growth Processes,” Nature 322 (1986), pp. 789–93; Richard Voss, “Random Fractal Forgeries: From Mountains to Music,” in Science and Uncertainty, ed. Sara Nash (London: IBM United Kingdom, 1985). CHARTED ON THE OLDER MAN’S BLACKBOARD Houthakker, Mandelbrot. WASSILY LEONTIEF Quoted in Fractal Geometry, p. 423. INTRODUCED FOR A LECTURE Woods Hole Oceanographic Institute, August 1985. BORN IN WARSAW Mandelbrot. BOURBAKI Mandelbrot, Richter. Little has been written about Bourbaki even now; one playful introduction is Paul R. Halmos, “Nicholas Bourbaki,” Scientific American 196 (1957), pp. 88–89. MATHEMATICS SHOULD BE SOMETHING Smale. THE FIELD DEVELOPS Peitgen. PIONEER-BY–NECESSITY “Second Stage,” p. 5. THIS HIGHLY ABSTRACT Mandelbrot; Fractal Geometry, p. 74; J.

THIS HIGHLY ABSTRACT Mandelbrot; Fractal Geometry, p. 74; J. M. Berger and Benoit Mandelbrot, “A New Model for the Clustering of Errors on Telephone Circuits,” IBM Journal of Research and Development 7 (1963), pp. 224–36. THE JOSEPH EFFECT Fractal Geometry, p. 248. CLOUDS ARE NOT SPHERES Ibid., p. 1, for example. WONDERING ABOUT COASTLINES Ibid., p. 27. THE PROCESS OF ABSTRACTION Ibid., p. 17. “THE NOTION” Ibid., p. 18. ONE WINTRY AFTERNOON Mandelbrot. THE EIFFEL TOWER Fractal Geometry, p. 131, and “On Fractal Geometry,” p. 1663. 102 ORIGINATED BY MATHEMATICIANS F. Hausdorff and A. S. Besicovich. “THERE WAS A LONG HIATUS” Mandelbrot. IN THE NORTHEASTERN Scholz; C. H. Scholz and C. A. Aviles, “The Fractal Geometry of Faults and Faulting,” preprint, Lamont-Doherty Geophysical Observatory; C. H. Scholz, “Scaling Laws for Large Earthquakes,” Bulletin of the Seismological Society of America 72 (1982), pp. 1–14.

pages: 338 words: 106,936

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


Albert Einstein, algorithmic trading, Antoine Gombaud: Chevalier de Méré, Asian financial crisis, bank run, beat the dealer, Benoit Mandelbrot, Black Swan, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, Brownian motion, butterfly effect, capital asset pricing model, Carmen Reinhart, Claude Shannon: information theory, collateralized debt obligation, collective bargaining, dark matter, Edward Lorenz: Chaos theory, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, George Akerlof, Gerolamo Cardano, Henri Poincaré, invisible hand, Isaac Newton, iterative process, John Nash: game theory, Kenneth Rogoff, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, martingale, Myron Scholes, new economy, Paul Lévy, Paul Samuelson, prediction markets, probability theory / Blaise Pascal / Pierre de Fermat, quantitative trading / quantitative finance, random walk, Renaissance Technologies, risk-adjusted returns, Robert Gordon, Robert Shiller, Robert Shiller, Ronald Coase, Sharpe ratio, short selling, Silicon Valley, South Sea Bubble, statistical arbitrage, statistical model, stochastic process, The Chicago School, The Myth of the Rational Market, tulip mania, V2 rocket, Vilfredo Pareto, volatility smile

“But then ‘a storm’ would come through . . .”: Mandelbrot describes this aspect of his wartime experience in Mandelbrot (1998). “This is a general property of fractals . . .”: There are many connections between fractals and fat-tailed distributions. That certain features of fractals exhibit fat tails is one such connection; another is that (some) fat-tailed distributions themselves exhibit self-similarity, in the form of power-law scaling in their tails. Mandelbrot was a central figure in identifying and exploring these relationships. See Mandelbrot (1997). “Known as the Butcher of Lyon . . .”: For more on Barbie, see Bower (1984) and McKale (2012). “. . . ‘there was no great distinction . . .’ ”: This quote is from Mandelbrot (1998). “. . . and economist named Vilfredo Pareto”: The definitive collection on Pareto and his influence is the three-volume Wood and McClure (1999); see also Cirillo (1979)

“It was one of his first attempts . . .”: Mandelbrot coined the term fractal in Mandelbrot (1975), which was translated into English as Mandelbrot (1977). But Mandelbrot (1967) is one of the first places where he describes geometrical objects with non-integer Hausdorff dimension exhibiting self-similarity. “. . . but anti-Semitism in the south was less virulent . . .”: While the comparative claim is true, it should not be taken to mean that anti-Semitism was not rampant in Vichy France. For more on Vichy France during World War II, including French anti-Semitism during the war, see Paxton (1972), Marrus and Paxton (1995), and Poznanski (2001). “. . . except to say that . . .”: These quotes come from the interview that Mandelbrot did for Web of Stories (Mandelbrot 1998). “In Thomas Pynchon’s novel Gravity’s Rainbow . . .”: This is Pynchon (1973).

Dismaying as it is for our Sisyphean surveyor, there’s no expected value for the average size of a feature on a coastline. This is a general property of fractals, following from their self-similarity. From one point of view, they are beautifully ordered and regular; from another, wildly random. And if fractals are everywhere, as Mandelbrot believed, the world is a place dominated by extremes, where our intuitive ideas about averages and normalcy can only lead us astray. Though he never provided details, Mandelbrot often alluded to a particularly harrowing experience toward the end of 1943, while he was hiding with members of the French resistance. Afterward, his protectors realized that Mandelbrot couldn’t remain in Tulle, and they secured a place for him as a postgraduate student at a preparatory school in Lyon. Moving Mandelbrot was a risky proposition. Lyon was one of the most dangerous cities in southern France for both Jews and resistance sympathizers; Mandelbrot was both.

pages: 210 words: 42,271

Programming HTML5 Applications by Zachary Kessin


barriers to entry, continuous integration, fault tolerance, Firefox, Google Chrome, mandelbrot fractal, QWERTY keyboard, web application, WebSocket

Worker communication Web Worker Fractal Example Example 8-1 is the “Hello World” of Web Workers. A more complex example is called for. Figure 8-3 shows a visual representation of a Mandelbrot set computed in a Web Worker. Here the worker and the main thread split up the work to draw the fractal. The worker does the actual work of computing the Mandelbrot set, while the frontend script takes that raw data and displays it in the canvas. Figure 8-3. Mandelbrot example The frontend script (see Example 8-2) sets up the canvas element and scales it to fit in the page. Then it creates an object to wrap the worker interface. The wrapper object creates the worker in the wrapper’s run() method, passing to the worker a parameter block that tells it what chunk of the Mandelbrot set to compute. The draw method takes the data, scales it to fit onto the canvas, sets a color, and then draws the pixel.

noglobals to, QUnit automatic validation test, Events blobs as, Blobs in manifest file, Structure of the Manifest File, Events preventing caching with query string, Debugging Manifest Files transforming files into, Working with Files web socket, The Web Sockets Interface user interface, Expanding Functions with Prototypes, Testing JavaScript Applications, Splitting Up Work Through Web Workers, Web Worker Fractal Example changing web worker behavior, Web Worker Fractal Example locking up, Splitting Up Work Through Web Workers processing input from, Expanding Functions with Prototypes test development for, Testing JavaScript Applications V variable(s), Closures, Closures, Closures, Closures, Array Iteration Operations, QUnit, A Simple Example, Selenese Command Programming Interface, IndexedDB, Web Worker Fractal Example, JavaScript Tools You Should Know binding, Closures, A Simple Example closures, Closures, Array Iteration Operations global, QUnit, Selenese Command Programming Interface, JavaScript Tools You Should Know in JavaScript, Closures, Closures mode, IndexedDB that local, Web Worker Fractal Example Venkman, JavaScript’s Triumph verifyElementPresent command (Selenium), Selenium Commands version control, IndexedDB, Adding Indexes, Introduction to the Manifest File, Updates to the Manifest File, Debugging Manifest Files <video> tag, Graphics, Audio and Video virtual machines, Selenium RC and a Test Farm W W3C IndexedDB draft proposal, IndexedDB WAI-ARIA, Accessibility Through WAI-ARIA waitFor statements, Selenium waitForElementNotPresent command (Selenium), Selenese Command Programming Interface waitForElementPresent command (Selenium), Selenium Commands, Selenese Command Programming Interface, Selenese Command Programming Interface Wampler, Dean, Functional Programming web fonts support, New CSS Web Inspector Remote, JavaScript’s Triumph web page drop zone example, Putting It All Together web server configuration files, Introduction to the Manifest File web sockets, Adding Power to Web Applications, Web Sockets, Web Socket Protocol Web Toolkit, Google, JavaScript’s Triumph Web Workers, Adding Power to Web Applications, Nonblocking I/O and Callbacks, IndexedDB, Splitting Up Work Through Web Workers, Graphics, Maps, Using Web Workers, The Worker Environment, Worker Communication, Web Worker Fractal Example, Web Worker Fractal Example, Testing and Debugging Web Workers, A Pattern for Reuse of Multithread Processing, A Pattern for Reuse of Multithread Processing, Libraries for Web Workers, Canvas and SVG communication example, Worker Communication creating, Using Web Workers environment, The Worker Environment and graphics, Graphics IndexedDB and, IndexedDB libraries for, Libraries for Web Workers Mandelbrot computation examples, Web Worker Fractal Example, Web Worker Fractal Example, Canvas and SVG and maps, Maps running functions with/without, A Pattern for Reuse of Multithread Processing, A Pattern for Reuse of Multithread Processing testing and debugging, Testing and Debugging Web Workers XHR objects in, Nonblocking I/O and Callbacks WebGL, Canvas and SVG WebKit, JavaScript’s Triumph, The localStorage and sessionStorage Objects WebKitBlobBuilder, Blobs webKitSlice() method (Chrome), Blobs weinre (Web Inspector Remote), JavaScript’s Triumph while loops, A Pattern for Reuse of Multithread Processing whitelisting resources, Debugging Manifest Files window.applicationCache object, Events windowMaximized command (Selenium), Selenium Commands X XMLHttpRequest (XHR), JavaScript’s Triumph, Nonblocking I/O and Callbacks, Functional Programming, Uploading Files, The Worker Environment XSS and injection attacks, IndexedDB Y Yahoo!

To allow the inner function to have access to that object, which it will need to draw a pixel, it is necessary to alias it to a lexically scoped variable. By convention that variable is often called that. Example 8-2. Mandelbrot frontend var drawMandelSet = function drawMandelSet(){ var mandelPanel = $('body'); var width = mandelPanel.innerWidth(); var height = mandelPanel.innerHeight(); var range = [{ x: -2, y: -1.4 }, { x: 5, y: 1.4 }]; $('canvas#fractal').height(height + 100); $('canvas#fractal').width(width - 50); var left = 0; var top = 0; var canvas = $("canvas#fractal")[0]; var ctx = canvas.getContext("2d"); var params = { range: range, startx: 0.0, starty: 0.0, width: width, height: height }; var y_array = []; var worker = { params: params, draw: function draw(data){ data.forEach(function d(point){ if (this.axis.x[point.drawLoc.x] === undefined) { this.axis.x[point.drawLoc.x] = point.point.x; } if (this.axis.y[height - point.drawLoc.y] === undefined) { this.axis.y[height - point.drawLoc.y] = point.point.y; } ctx.fillStyle = pickColor(point.escapeValue); ctx.fillRect(point.drawLoc.x + 0.5, height - point.drawLoc.y + 0.5, 1, 1); }, this); }, axis: { x: [], y: [], find: function(x, y){ return new Complex(this.x[x], this.y[y]); }, reset: function(){ this.x = [], this.y = []; } }, myWorker: false, run: function startWorker(params){ this.myWorker = new Worker("js/worker.js"); var that = this; this.myWorker.postMessage(JSON.stringify(params)); this.myWorker.onmessage = function(event){ var data = JSON.parse(; if (data.type === 'draw') { that.draw(JSON.parse(; } else if ( === 'log') {; } }; } };; return worker; }; $(document).ready(drawMandelSet); Function.prototype.createDelegate = function createDelegate(scope){ var fn = this; return function(){, arguments); }; }; function pickColor(escapeValue){ if (escapeValue === Complex.prototype.max_iteration) { return "black"; } var tone = 255 - escapeValue * 10; var colorCss = "rgb({r},{g},{b})".populate({ r: tone, g: tone, b: tone }); return colorCss; } String.prototype.populate = function populate(params) { var str = this.replace(/\{\w+\}/g, function stringFormatInner(word) { return params[word.substr(1, word.length - 2)]; }); return str; }; The actual worker (see Example 8-3) is very simple.

pages: 578 words: 168,350

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

Alfred Russel Wallace, Anton Chekhov, Benoit Mandelbrot, Black Swan, British Empire, butterfly effect, carbon footprint, Cesare Marchetti: Marchetti’s constant, clean water, complexity theory, computer age, conceptual framework, continuous integration, corporate social responsibility, correlation does not imply causation, creative destruction, dark matter, Deng Xiaoping, double helix, Edward Glaeser, endogenous growth, Ernest Rutherford, first square of the chessboard, first square of the chessboard / second half of the chessboard, Frank Gehry, Geoffrey West, Santa Fe Institute, Guggenheim Bilbao, housing crisis, Index librorum prohibitorum, invention of agriculture, invention of the telephone, Isaac Newton, Jane Jacobs, Jeff Bezos, Johann Wolfgang von Goethe, John von Neumann, Kenneth Arrow, laissez-faire capitalism, life extension, Mahatma Gandhi, mandelbrot fractal, Marchetti’s constant, Masdar, megacity, Murano, Venice glass, Murray Gell-Mann, New Urbanism, Peter Thiel, profit motive, publish or perish, Ray Kurzweil, Richard Feynman, Richard Feynman, Richard Florida, Silicon Valley, smart cities, Stephen Hawking, Steve Jobs, Stewart Brand, technological singularity, The Coming Technological Singularity, The Death and Life of Great American Cities, the scientific method, too big to fail, transaction costs, urban planning, urban renewal, Vernor Vinge, Vilfredo Pareto, Von Neumann architecture, Whole Earth Catalog, Whole Earth Review, wikimedia commons, working poor

Just as we need to know if it is in miles, centimeters, or angstroms, we also need to know the resolution that was used. Mandelbrot introduced the concept of a fractal dimension, defined by adding 1 to the exponent of the power law (the value of the slopes). Thus the fractal dimension of the South African coast is 1.02, Norway 1.52, and so on. The point of adding the 1 was to connect the idea of fractals to the conventional concept of ordinary dimensions discussed in chapter 2. Recall that a smooth line has dimension 1, a smooth surface dimension 2, and a volume dimension 3. Thus the South African coast is very close to being a smooth line because its fractal dimension is 1.02, which is very close to 1, whereas Norway is far from it because its fractal dimension of 1.52 is so much greater than 1. You could imagine an extreme case of this in which the line is so crinkly and convoluted that it effectively fills an entire area.

Benoit Mandelbrot, “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,” Science 156 (1967): 636–38. 23. See, for example, Rosario N. Mantegna and H. Eugene Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge, UK: Cambridge University Press, 1999). 24. See, for example, J. B. Bassingthwaighte, L. S. Liebovitch, and B. J. West, Fractal Physiology (New York: Oxford University Press, 1994). 25. Mandelbrot, The Fractal Geometry of Nature. 26. See, for instance, Manfred Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (New York: W. H. Freeman, 1991). 4. THE FOURTH DIMENSION OF LIFE 1. G. B. West, J. H. Brown, and B. J. Enquist, “The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms,” Science 284 (1999): 1677–79. 2.

In 1982 Mandelbrot published a highly influential and very readable semipopular book titled The Fractal Geometry of Nature.25 This inspired tremendous interest in fractals by showing their ubiquity across both science and the natural world. It stimulated a mini industry searching for fractals, finding them everywhere, measuring their dimensions, and showing how their magical properties result in wonderfully exotic geometric figures. Mandelbrot had shown how relatively simple algorithmic rules based on fractal mathematics can produce surprisingly complex patterns. He, and later many others, produced amazingly realistic simulations of mountain ranges and landscapes, as well as intriguing psychedelic patterns. This was enthusiastically embraced by the film and media industries, so much so that a great deal of what you now see on the screen and in advertisements, whether “realistic” battle scenes, glorious landscapes, or futuristic fantasy, is based on fractal paradigms.

pages: 197 words: 35,256

NumPy Cookbook by Ivan Idris


business intelligence, cloud computing, computer vision, Debian,, Eratosthenes, mandelbrot fractal, p-value, sorting algorithm, statistical model, transaction costs, web application

For more information see the Wikipedia article already mentioned in this recipe: for n in range(ITERATIONS): print n mask = numpy.abs(z) <= 4 z[mask] = z[mask] ** 2 + c[mask] fractal[(fractal == MAX_COLOR) & (-mask)] = (MAX_COLOR - 1) * n / ITERATIONS Plot the fractal.Plot the fractal with Matplotlib: matplotlib.pyplot.subplot(211) matplotlib.pyplot.imshow(fractal) matplotlib.pyplot.title('Mandelbrot') matplotlib.pyplot.axis('off') Combine the fractal and Lena.Use the choose function to pick value from the fractal or Lena array: matplotlib.pyplot.subplot(212) matplotlib.pyplot.imshow(numpy.choose(fractal < lena, [fractal, lena])) matplotlib.pyplot.axis('off') matplotlib.pyplot.title('Mandelbrot + Lena') The following is the resulting image: The following is the complete code for this recipe: import numpy import matplotlib.pyplot import sys import scipy if(len(sys.argv) !

For more information see the Wikipedia article already mentioned in this recipe: for n in range(ITERATIONS): print n mask = numpy.abs(z) <= 4 z[mask] = z[mask] ** 2 + c[mask] fractal[(fractal == MAX_COLOR) & (-mask)] = (MAX_COLOR - 1) * n / ITERATIONS Plot the fractal.Plot the fractal with Matplotlib: matplotlib.pyplot.subplot(211) matplotlib.pyplot.imshow(fractal) matplotlib.pyplot.title('Mandelbrot') matplotlib.pyplot.axis('off') Combine the fractal and Lena.Use the choose function to pick value from the fractal or Lena array: matplotlib.pyplot.subplot(212) matplotlib.pyplot.imshow(numpy.choose(fractal < lena, [fractal, lena])) matplotlib.pyplot.axis('off') matplotlib.pyplot.title('Mandelbrot + Lena') The following is the resulting image: The following is the complete code for this recipe: import numpy import matplotlib.pyplot import sys import scipy if(len(sys.argv) != 2): print "Please input the number of iterations for the fractal" sys.exit() ITERATIONS = int(sys.argv[1]) lena = scipy.misc.lena() SIZE = lena.shape[0] MAX_COLOR = 255. x_min, x_max = -2.5, 1 y_min, y_max = -1, 1 # Initialize arrays x, y = numpy.meshgrid(numpy.linspace(x_min, x_max, SIZE), numpy.linspace(y_min, y_max, SIZE)) c = x + 1j * y z = c.copy() fractal = numpy.zeros(z.shape, dtype=numpy.uint8) + MAX_COLOR # Generate fractal for n in range(ITERATIONS): print n mask = numpy.abs(z) <= 4 z[mask] = z[mask] ** 2 + c[mask] fractal[(fractal == MAX_COLOR) & (-mask)] = (MAX_COLOR - 1) * n / ITERATIONS # Display the fractal matplotlib.pyplot.subplot(211) matplotlib.pyplot.imshow(fractal) matplotlib.pyplot.title('Mandelbrot') matplotlib.pyplot.axis('off') # Combine with lena matplotlib.pyplot.subplot(212) matplotlib.pyplot.imshow(numpy.choose(fractal < lena, [fractal, lena])) matplotlib.pyplot.axis('off') matplotlib.pyplot.title('Mandelbrot + Lena') How it works...

See also The Installing Matplotlib recipe in Chapter 1, Winding Along with IPython Combining images In this recipe, we will combine the famous Mandelbrot fractal (for more information on Madelbrot set visit and the image of Lena. These types of fractals are defined by a recursive formula, where you calculate the next complex number in a series by multiplying the current complex number you have, by itself and adding a constant to it. Getting ready Install SciPy, if necessary. The See Also section of this recipe, has a reference to the related recipe. How to do it... We will start by initializing the arrays, followed by generating and plotting the fractal, and finally, combining the fractal with the Lena image. Initialize the arrays.We will initialize x, y, and z arrays corresponding to the pixels in the image area with the meshgrid, zeros, and linspace functions: x, y = numpy.meshgrid(numpy.linspace(x_min, x_max, SIZE), numpy.linspace(y_min, y_max, SIZE)) c = x + 1j * y z = c.copy() fractal = numpy.zeros(z.shape, dtype=numpy.uint8) + MAX_COLOR Generate the fractal.If z is a complex number, you have the following relation for a Mandelbrot fractal: In this equation, c is a constant complex number.

pages: 257 words: 13,443

Statistical Arbitrage: Algorithmic Trading Insights and Techniques by Andrew Pole


algorithmic trading, Benoit Mandelbrot, Chance favours the prepared mind, constrained optimization, Dava Sobel, George Santayana, Long Term Capital Management, Louis Pasteur, mandelbrot fractal, market clearing, market fundamentalism, merger arbitrage, pattern recognition, price discrimination, profit maximization, quantitative trading / quantitative finance, risk tolerance, Sharpe ratio, statistical arbitrage, statistical model, stochastic volatility, systematic trading, transaction costs

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. It is unknown whether any successful trading strategies have been built using fractal analysis; Mandelbrot himself does not believe his tools are yet sufficiently developed for prediction of financial series to be feasible. 3.8 PRACTICAL MATTERS Forecasts of stock price movements are incredibly inaccurate.

Norton & Co., 1993. Institutional Investor. ‘‘Wall Street South,’’ Institutional Investor, March 2004. Johnson, N.L., S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions, Volumes I and II. New York: John Wiley & Sons, 1994. Lehman Brothers. Algorithmic Trading. New York: Lehman Brothers, 2004. Mandelbrot, B.B. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. New York: Springer-Verlag, 1997. Mandelbrot B.B., and R.L. Hudson. The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books, 2004. Orwell, George. 1984. New York: New American Library, 1950. Perold, A.F. (1988). ‘‘The Implementation Shortfall, Paper vs. Reality,’’ Journal of Portfolio Management, 14:3, 4–9. Pole, A., M. West, and J. Harrison. Applied Bayesian Forecasting and Time Series Analysis.

The maximum value this probability can assume is 12 when p = 12 (differentiate, equate to zero, solve). 4.2.5 Generalizations Financial time series are notorious for the tenacity with which they refuse to reveal underlying mathematical structure (though Mandelbrot, 2004, may demur from that statement). Features of such data, which often show up in statistical modeling, include: nonnormal distributions (returns are frequently characterized by leptokurtosis); nonconstant variance (market dynamics often produce bursts of high and low volatility, and modelers have tried many approaches from GARCH and its variants to Mandelbrot’s fractals, see Chapter 3); and serial dependence. The conditions of the theorem can be relaxed to accommodate all of these behaviors. 74 STATISTICAL ARBITRAGE The result extends to arbitrary continuous random variables directly: The constraint of support on the nonnegative real line is not required.

pages: 295 words: 66,824

A Mathematician Plays the Stock Market by John Allen Paulos


Benoit Mandelbrot, Black-Scholes formula, Brownian motion, business climate, butterfly effect, capital asset pricing model, correlation coefficient, correlation does not imply causation, Daniel Kahneman / Amos Tversky, diversified portfolio, Donald Trump, double entry bookkeeping, Elliott wave, endowment effect, Erdős number, Eugene Fama: efficient market hypothesis, four colour theorem, George Gilder, global village, greed is good, index fund, intangible asset, invisible hand, Isaac Newton, John Nash: game theory, Long Term Capital Management, loss aversion, Louis Bachelier, mandelbrot fractal, margin call, mental accounting, Myron Scholes, Nash equilibrium, Network effects, passive investing, Paul Erdős, Paul Samuelson, Ponzi scheme, price anchoring, Ralph Nelson Elliott, random walk, Richard Thaler, Robert Shiller, Robert Shiller, short selling, six sigma, Stephen Hawking, survivorship bias, transaction costs, ultimatum game, Vanguard fund, Yogi Berra

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. And the bottom line, or, in this case, the bottom fractal, for stocks? By starting with the basic up-down-up and down-up-down patterns of a stock’s possible movements, continually replacing each of these patterns’ three segments with smaller versions of one of the basic patterns chosen at random, and then altering the spikiness of the patterns to reflect changes in the stock’s volatility, Mandelbrot has constructed what he calls multifractal “forgeries.”

Most commonly the market rises in five distinct waves and declines in three distinct waves for obscure psychological or systemic reasons. 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. To save the day, complications are introduced into the theory, so many, in fact, that the theory soon becomes incapable of being falsified.

This non-predictability is the result not of randomness but of complexity too great to fathom. Yet another reason to suspect that parts of the market may be better modeled by nonlinear systems is that such systems’ “trajectories” often follow a fractal course. The trajectories of these systems, of which the stock price movements may be considered a proxy, turn out to be aperiodic and unpredictable and, when examined closely, evince even more intricacy. Still closer inspection of the system’s trajectories reveals yet smaller vortices and complications of the same general kind. In general, fractals are curves, surfaces, or higher dimensional objects that contain more, but similar, complexity the closer one looks. A shoreline, to cite a classic example, has a characteristic jagged shape at whatever scale we draw it; that is, whether we use satellite photos to sketch the whole coast, map it on a fine scale by walking along some small section of it, or examine a few inches of it through a magnifying glass.

pages: 442 words: 39,064

Why Stock Markets Crash: Critical Events in Complex Financial Systems by Didier Sornette


Asian financial crisis, asset allocation, Berlin Wall, Bretton Woods, Brownian motion, capital asset pricing model, capital controls, continuous double auction, currency peg, Deng Xiaoping, discrete time, diversified portfolio, Elliott wave, Erdős number, experimental economics, financial innovation, floating exchange rates, frictionless, frictionless market, full employment, global village, implied volatility, index fund, information asymmetry, intangible asset, invisible hand, John von Neumann, joint-stock company, law of one price, Louis Bachelier, mandelbrot fractal, margin call, market bubble, market clearing, market design, market fundamentalism, mental accounting, moral hazard, Network effects, new economy, oil shock, open economy, pattern recognition, Paul Erdős, Paul Samuelson, quantitative trading / quantitative finance, random walk, risk/return, Ronald Reagan, Schrödinger's Cat, selection bias, short selling, Silicon Valley, South Sea Bubble, statistical model, stochastic process, Tacoma Narrows Bridge, technological singularity, The Coming Technological Singularity, The Wealth of Nations by Adam Smith, Tobin tax, total factor productivity, transaction costs, tulip mania, VA Linux, Y2K, yield curve

In other words, the hierarchical diamond and tree networks have the property of reproducing themselves exactly on different magnifications. Such a property has been coined “fractal” by Mandelbrot [284], who recognized, based on the pioneering work of Richardson [343], that many natural and social phenomena are endowed, at least approximately, with the scale invariance symmetry. Many of us have met fractals through their beautiful, delicately complex pictures, which are usually computer generated. Modern Hollywood movies use landscapes, mountain ranges, cloud structures, and other artificial constructions that are computer generated according to recipes devised to obtain fractal geometries. It turns out that many of the natural structures of the world are approximately fractal [29, 126, 88, 31, 292, 394] and that our aesthetic sense resonates with fractal forms. In most simple fractal constructions and textbook examples, the scale invariance property does not hold for arbitrary magnification.

To capture this novel concept, we already mentioned that the word “fractal” was coined by Mandelbrot [284], from the Latin root fractus to capture the rough, broken, and irregular characteristics of the objects presenting at least approximately the property of scale invariance. This roughness can be present at all scales, which distinguishes fractals from Euclidean shapes. Mandelbrot worked actively to demonstrate that this concept is not just a mathematical curiosity but has strong relevance to the real world. The remarkable fact is that this generalization, from integer dimensions to fractional dimensions, has a profound and intuitive interpretation: noninteger dimensions describe irregular sets consisting of parts similar to the whole. There are many examples of (approximate) fractals in nature, such as the distribution of galaxies at large scales, certain mountain ranges, fault networks and earthquake locations, rocks, lightning bolts, snowflakes, hier archies and l o g - p e r i o d i c i t y 189 Fig. 6.9.

., Morein, G., and Turcotte, D. L. (1998). Forest fires: An example of self-organized critical behavior, Science 281, 1840–1842. 281. Malki, E. (1999). The Financial Crisis in Russia, ewp-mac/9901001. 282. Malkiel, B. G. (1999). A Random Walk Down Wall Street (Norton, New York). 283. Mandelbrot, B. B. (1967). How long is the coast of Britain? Statistical selfsimilarity and fractional dimension, Science 155, 636–638. 284. Mandelbrot, B. B. (1982). The Fractal Geometry of Nature (Freeman, San Francisco). 285. Mandelbrot, B. B. (1999). A multifractal walk down Wall Street, Scientific American 280, 70–73(February). 286. Mantegna, R. N., Buldyrev, S. V., Goldberger, A. L., Halvin, S., and Stanley, H. E. (1995). Systematic analysis of coding and non-coding sequences using methods of statistical linguistics, Physical Review E 52, 2939–2950. 287.

pages: 651 words: 180,162

Antifragile: Things That Gain From Disorder by Nassim Nicholas Taleb


Air France Flight 447, Andrei Shleifer, banking crisis, Benoit Mandelbrot, Berlin Wall, Black Swan, Chuck Templeton: OpenTable, commoditize, creative destruction, credit crunch, Daniel Kahneman / Amos Tversky, David Ricardo: comparative advantage, discrete time, double entry bookkeeping, Emanuel Derman, epigenetics, financial independence, Flash crash, Gary Taubes, George Santayana, Gini coefficient, Henri Poincaré, high net worth, hygiene hypothesis, Ignaz Semmelweis: hand washing, informal economy, invention of the wheel, invisible hand, Isaac Newton, James Hargreaves, Jane Jacobs, joint-stock company, joint-stock limited liability company, Joseph Schumpeter, Kenneth Arrow, knowledge economy, Lao Tzu, Long Term Capital Management, loss aversion, Louis Pasteur, mandelbrot fractal, Marc Andreessen, meta analysis, meta-analysis, microbiome, money market fund, moral hazard, mouse model, Myron Scholes, Norbert Wiener, pattern recognition, Paul Samuelson, placebo effect, Ponzi scheme, principal–agent problem, purchasing power parity, quantitative trading / quantitative finance, Ralph Nader, random walk, Ray Kurzweil, rent control, Republic of Letters, Ronald Reagan, Rory Sutherland, selection bias, Silicon Valley, six sigma, spinning jenny, statistical model, Steve Jobs, Steven Pinker, Stewart Brand, stochastic process, stochastic volatility, The Great Moderation, the new new thing, The Wealth of Nations by Adam Smith, Thomas Bayes, Thomas Malthus, too big to fail, transaction costs, urban planning, Vilfredo Pareto, Yogi Berra, Zipf's Law

Further, things that grow in a natural way, whether cities or individual houses, have a fractal quality to them. Like everything alive, all organisms, like lungs, or trees, grow in some form of self-guided but tame randomness. What is fractal? Recall Mandelbrot’s insight in Chapter 3: “fractal” entails both jaggedness and a form of self-similarity in things (Mandelbrot preferred “self-affinity”), such as trees spreading into branches that look like small trees, and smaller and smaller branches that look like a slightly modified, but recognizable, version of the whole. These fractals induce a certain wealth of detail based on a small number of rules of repetition of nested patterns. The fractal require some jaggedness, but one that has some method to its madness. Everything in nature is fractal, jagged, and rich in detail, though with a certain pattern.

Journal of Global Information Management 8(3): 5. Malmendier, U., and G. Tate, 2008, “Who Makes Acquisitions? CEO Overconfidence and the Market’s Reaction.” Journal of Financial Economics 89(1): 20–43. Malmendier, U., and G. Tate, 2009, “Superstar CEOs.” Quarterly Journal of Economics 124(4): 1593–1638. Mandelbrot, Benoît B., 1983, The Fractal Geometry of Nature. W. H. Freeman. Mandelbrot, Benoît B., 1997, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. New York: Springer-Verlag. Mandelbrot, Benoît B., and N. N. Taleb, 2010, “Random Jump, Not Random Walk.” In Richard Herring, ed., The Known, the Unknown, and the Unknowable. Princeton, N.J.: Princeton University Press. Mansel, P., 2012, Levant. Hachette. Marglin, S. A., 1996, “Farmers, Seedsmen, and Scientists: Systems of Agriculture and Systems of Knowledge.”

Further, much French architecture was a response to the tax on windows and doors installed after the Revolution, so many buildings have a very small number of windows. Just as with the unintrusive shoes that allow us to feel the terrain, modern technology allows some of us to reverse that trend, as expressed by Oswald Spengler, which makes civilization go from plants to stone, that is, from the fractal to the Euclidian. We are now moving back from the smooth stone to the rich fractal and natural. Benoît Mandelbrot wrote in front of a window overlooking trees: he craved fractal aesthetics so much that the alternative would have been inconceivable. Now modern technology allows us to merge with nature, and instead of a small window, an entire wall can be transparent and face lush and densely forested areas. Metrification One example of the neomania of states: the campaign for metrification, that is, the use of the metric system to replace “archaic” ones on grounds of efficiency—it “makes sense.”

pages: 461 words: 128,421

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


activist fund / activist shareholder / activist investor, Albert Einstein, Andrei Shleifer, asset allocation, asset-backed security, bank run, beat the dealer, Benoit Mandelbrot, Black-Scholes formula, Bretton Woods, Brownian motion, capital asset pricing model, card file, Cass Sunstein, collateralized debt obligation, complexity theory, corporate governance, corporate raider, Credit Default Swap, credit default swaps / collateralized debt obligations, Daniel Kahneman / Amos Tversky, David Ricardo: comparative advantage, discovery of the americas, diversification, diversified portfolio, Edward Glaeser, Edward Thorp, endowment effect, Eugene Fama: efficient market hypothesis, experimental economics, financial innovation, Financial Instability Hypothesis, fixed income, floating exchange rates, George Akerlof, Henri Poincaré, Hyman Minsky, implied volatility, impulse control, index arbitrage, index card, index fund, information asymmetry, invisible hand, Isaac Newton, John Meriwether, John Nash: game theory, John von Neumann, joint-stock company, Joseph Schumpeter, Kenneth Arrow, libertarian paternalism, linear programming, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, market bubble, market design, Myron Scholes, New Journalism, Nikolai Kondratiev, Paul Lévy, Paul Samuelson, pension reform, performance metric, Ponzi scheme, prediction markets, pushing on a string, quantitative trading / quantitative finance, Ralph Nader, RAND corporation, random walk, Richard Thaler, risk/return, road to serfdom, Robert Bork, Robert Shiller, Robert Shiller, rolodex, Ronald Reagan, shareholder value, Sharpe ratio, short selling, side project, Silicon Valley, South Sea Bubble, statistical model, The Chicago School, The Myth of the Rational Market, The Predators' Ball, the scientific method, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, Thomas Kuhn: the structure of scientific revolutions, Thomas L Friedman, Thorstein Veblen, Tobin tax, transaction costs, tulip mania, value at risk, Vanguard fund, Vilfredo Pareto, volatility smile, Yogi Berra

That clearly wasn’t the case, and the search for better models of volatility was now on in earnest. One starting point was the statistical framework assembled twenty-five years before by Benoit Mandelbrot. Mandelbrot hadn’t predicted black Monday. He hadn’t written anything about finance in years. But anyone who had studied his market writings from the 1960s was far less surprised by events on Wall Street than those who had restricted their reading to standard finance textbooks. Mandelbrot was by this time also becoming famous. His reputation-making Fractal Geometry of Nature came out in 1982. The year of the crash, journalist James Gleick’s bestselling book Chaos introduced him to a much wider readership. After 1987, Mandelbrot’s long-ignored ideas began to intrude upon the consciousness of Wall Street. It wasn’t so much his probability formulas that caught on—he had only written them down, he said later, because at the time “random processes could only be investigated through formulas and theorems.”

These patterns, which allow far more room for outliers than the standard bell curve, had first been observed around the turn of the nineteenth century in the distribution of wealth,2 and it was the statistics of wealth and income that Mandelbrot studied. Then he visited Hendrik Houthakker’s Harvard classroom and saw that cotton futures prices fell into the same pattern as incomes and words. It wasn’t just the ski jump line; the data was also “self-similar”—that is, charts of small snippets looked just like those of large swaths. Mandelbrot was later to find similar patterns in historical climate data along the Nile, the coast of Britain, and the ins and outs of tree bark. After he dubbed them “fractals” in 1982, he was hailed as a visionary, one of the progenitors of the new science of chaos and complexity that was transforming physics and other fields. By then, though, Mandelbrot had long abandoned finance. At the beginning he had been warmly welcomed into the small but growing fellowship of random walkers.

Paul Samuelson, “Coping Sensibly,” Newsweek, March 6, 1978, 88. 42. I’m referring mainly to the account in Roger Lowenstein’s Buffett: The Making of an American Capitalist. 43. “A Conversation With Benjamin Graham,” Financial Analysts Journal (Sept./Oct. 1976): 20–23. CHAPTER 8: FISCHER BLACK CHOOSES TO FOCUS ON THE PROBABLE 1. Mandelbrot tells the story of encountering Zipf’s work in Benoit Mandelbrot and Richard L. Hudson, The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward (New York: Basic Books, 2004), 150–59. The Zipf book mentioned is Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology (Cambridge, Mass.: Addison-Wesley, 1949). 2. This field had been pioneered by Italian mathematical economist Vilfredo Pareto. Pareto made important contributions to equilibrium economics and Irving Fisher visited him during his European grand tour in 1894.

pages: 381 words: 101,559

Currency Wars: The Making of the Next Gobal Crisis by James Rickards


Asian financial crisis, bank run, Benoit Mandelbrot, Berlin Wall, Big bang: deregulation of the City of London, Black Swan, borderless world, Bretton Woods, BRICs, British Empire, business climate, capital controls, Carmen Reinhart, Cass Sunstein, collateralized debt obligation, complexity theory, corporate governance, Credit Default Swap, credit default swaps / collateralized debt obligations, currency manipulation / currency intervention, currency peg, Daniel Kahneman / Amos Tversky, Deng Xiaoping, diversification, diversified portfolio, Fall of the Berlin Wall, family office, financial innovation, floating exchange rates, full employment, game design, German hyperinflation, Gini coefficient, global rebalancing, global reserve currency, high net worth, income inequality, interest rate derivative, John Meriwether, Kenneth Rogoff, labour mobility, laissez-faire capitalism, liquidity trap, Long Term Capital Management, mandelbrot fractal, margin call, market bubble, Mexican peso crisis / tequila crisis, money market fund, money: store of value / unit of account / medium of exchange, Myron Scholes, Network effects, New Journalism, Nixon shock, offshore financial centre, oil shock, one-China policy, open economy, paradox of thrift, Paul Samuelson, price mechanism, price stability, private sector deleveraging, quantitative easing, race to the bottom, RAND corporation, rent-seeking, reserve currency, Ronald Reagan, sovereign wealth fund, special drawing rights, special economic zone, The Myth of the Rational Market, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, Thomas Kuhn: the structure of scientific revolutions, time value of money, too big to fail, value at risk, War on Poverty, Washington Consensus, zero-sum game

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.” A line has one dimension (length) and a square has two dimensions (length and width). A fractal dimension of 1½ is something in between. A familiar example is the ubiquitous stock market chart of the kind shown in daily papers and financial websites. The chart itself consists of more than a single line (it has hundreds of small lines) but is less than an entire square (there is lots of unfilled space away from the lines).

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. Chaisson, Cosmic Evolution: The Rise of Complexity in Nature, Cambridge: Harvard University Press, 2001. Chaisson’s specific values for free energy rate densities are given as: 219 In his most ambitious work . . .

Unrestricted Warfare Symposium, three volumes. Laurel, MD: Johns Hopkins University Applied Physics Laboratory, 2007–2009. MacMillan, Margaret. Paris 1919: Six Months That Changed the World. New York: Random House, 2001. Makin, John H. The Global Debt Crisis: America’s Growing Involvement. New York: Basic Books, 1984. Mallaby, Sebastian. More Money Than God. New York: Penguin, 2010. Mandelbrot, Benoît, and Richard L. Hudson. The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books, 2004. Mead, Walter Russell. God and Gold: Britain, America, and the Making of the Modern World. New York: Random House, 2007. Meltzer, Allan H. A History of the Federal Reserve, Volume 1: 1913–1951. Chicago: University of Chicago Press, 2003. Mihm, Stephen. A Nation of Counterfeiters: Capitalism, Con Men, and the Making of the United States.

pages: 222 words: 53,317

Overcomplicated: Technology at the Limits of Comprehension by Samuel Arbesman


3D printing, algorithmic trading, Anton Chekhov, Apple II, Benoit Mandelbrot, citation needed, combinatorial explosion, Danny Hillis, David Brooks, digital map, discovery of the americas,, Erik Brynjolfsson, Flash crash, friendly AI, game design, Google X / Alphabet X, Googley, HyperCard, Inbox Zero, Isaac Newton, iterative process, Kevin Kelly, Machine translation of "The spirit is willing, but the flesh is weak." to Russian and back, mandelbrot fractal, Minecraft, Netflix Prize, Nicholas Carr, Parkinson's law, Ray Kurzweil, recommendation engine, Richard Feynman, Richard Feynman, Richard Feynman: Challenger O-ring, Second Machine Age, self-driving car, software studies, statistical model, Steve Jobs, Steve Wozniak, Steven Pinker, Stewart Brand, superintelligent machines, Therac-25, Tyler Cowen: Great Stagnation, urban planning, Watson beat the top human players on Jeopardy!, Whole Earth Catalog, Y2K

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. researchers analyzed the United States Code: William Li et al., “Law Is Code: A Software Engineering Approach to Analyzing the United States Code,” Journal of Business and Technology Law 10, no. 2 (2015): 297–372.

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.” So, too, our technological systems, once embedded in the real world, are far from the cleanly pristine logical constructions of the drawing board; they are full of the miscellaneous details of biology that have accreted over time, much like the evolutionary hodgepodge found within living systems.

The law professor David Post and the biologist Michael Eisen teamed up to examine this as well, and while they admit they can’t prove that a legal statement can always branch further, and that it’s “turtles all the way down,” they do note that “we have never met a legal question that could not be decomposed into subquestions.” Post and Eisen even show through simulations that certain types of branching structures that mimic legal systems actually can have a fractal structure. Testing this, they find features indicative of fractals when looking at actual legal citations of court case opinions. The fractal complexity of the law might be more than an evocative metaphor. As the scholars Mark Flood and Oliver Goodenough recognize, “Much of the value of good contracts and good lawyering derives from the seemingly tedious planning for all the ways that a relationship might run off the rails.” In other words, legal complexity is often derived from exceptions and their complications.

pages: 247 words: 43,430

Think Complexity by Allen B. Downey


Benoit Mandelbrot, cellular automata, Conway's Game of Life, Craig Reynolds: boids flock, discrete time,, Frank Gehry, Gini coefficient, Guggenheim Bilbao, mandelbrot fractal, Occupy movement, Paul Erdős, peer-to-peer, Pierre-Simon Laplace, sorting algorithm, stochastic process, strong AI, Thomas Kuhn: the structure of scientific revolutions, Turing complete, Turing machine, Vilfredo Pareto, We are the 99%

In their 2004 paper, “Efficient algorithm for the forest fire model,” they present evidence that the system is not critical after all ( 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. To support this argument, Mandelbrot points out the following: The distribution of observed data is often “the joint effect of a fixed underlying ‘true distribution’ and a highly variable ‘filter.’” Long-tailed distributions are robust to filtering; that is, “a wide variety of filters leave their asymptotic behavior unchanged.”

, Explanatory Models exponent, Order of Growth exponential distribution, Continuous Distributions exponential growth, Order of Growth extend, Summing Lists F falsifiability, Falsifiability Fast Fourier Transform (FFT), Fast Fourier Transform FIFO queue, FIFO Implementation fireflies, Paradigm Shift? flock behavior, Boids for loop, Iterators forest, What’s a Graph? forest fire model, Fractal CAs, Percolation, Reductionism and Holism four-color theorem, The Axes of Scientific Models Fourier transform, Spectral Density fractal, Fractal CAs fractal cellular automaton, Fractal CAs fractal dimension, Fractals, Sand Piles fractal geometry, Sand Piles The Fractal Geometry of Nature, Reductionism and Holism fractals, Fractals free will, A New Kind of Thinking, Determinism frequency, Zipf’s Law, Spectral Density frequentist probability, A New Kind of Thinking Freud, Sigmund, Instrumentalism fringe science, Paradigm Shift? Fuzzy Thinking, A New Kind of Thinking G Game of Life, Game of Life, Life Patterns, Pink Noise patterns, Life Patterns game theory, Prisoner’s Dilemma Gardner, Martin, Game of Life gather operator, A New Kind of Thinking Gaussian distribution, Continuous Distributions Gehry, Frank, A New Kind of Engineering generative model, Watts and Strogatz, Barabási and Albert generator, Generators, Hashtables genes, Realism, Reductionism and Holism genetics, Reductionism and Holism geometric objects, Fractals geometric resizing, Hashtables ghost cells, CADrawer glider, Life Patterns glider gun, Conway’s Conjecture Gödel’s Incompleteness Theorem, A New Kind of Thinking Goldstein, Rebecca, A New Kind of Thinking Gosper, Bill, Conway’s Conjecture Graph class, Representing Graphs graph algorithms, What Is This Book About?

Figure 8-3 shows what it looks like after 64 steps, and Figure 8-4 shows versus on a log-log scale. To estimate , I fit a line to this curve; its slope is 1.56. is a non-integer, which means that this set of points is a fractal. As t increases, the slope approaches , which is the fractal dimension of Sierpiński’s triangle. See Example 8-1. Write a function that takes a CA object; plots versus , where ; and estimates . Can you find other CAs with non-integer fractal dimensions? Be careful, you might have to run the CA for a while before converges. Here are some functions from numpy you might find useful: cumsum, log, and polyfit. You can download my solution from Example 8-2. In 1990, Bak, Chen, and Tang proposed a cellular automaton that is an abstract model of a forest fire. Each cell is in one of three states: empty, occupied by a tree, or on fire.

pages: 398 words: 31,161

Gnuplot in Action: Understanding Data With Graphs by Philipp Janert


bioinformatics, business intelligence, centre right, Debian, general-purpose programming language, iterative process, mandelbrot fractal, pattern recognition, random walk, Richard Stallman, six sigma, survivorship bias

In contrast, the long interval between 1,500 and 10,000 iteration steps is colored in a uniform white, because there are only few pixels in the image falling into this region (mostly the thin white boundaries that you can see around the solid black regions, which belong to the interior or the Mandelbrot set). There’s no reason to waste visual gradients on parameter ranges that occupy only a small and not very relevant area of the plot. Grayscale is of course only the first step. In color figure 5, I show the same data set, plotted with two different palettes, which are also listed in listing 9.3. One is huebased; the other is luminance-based. According to our guidelines, a luminance-based 6 This isn’t the place to give a detailed introduction to fractals and the Mandelbrot set. Plenty of information is readily available on the Internet—the Wikipedia entry for the Mandelbrot set is a good place to start. 172 CHAPTER 9 Color Figure 9.3 A black-and-white rendition of a section of the Mandelbrot set. Note the strongly uneven grayscale, visible in the colorbox.

See exploratory data analysis edge effects 282 enabling interlacing 207 multiplot mode 176 polar mode 184 encoding option 322 enhanced text mode 202 control characters 203 examples 204 options 339 enhancing quantitative perception 289 enum type indicator xxviii environment variables 194, 234 EPS file 211 epslatex terminals 212 error integral functions 317 errorbars 78 Euclidean distance 150 European Union 262 every directive 31 example data sets airplane 301 allometric scaling 254 armor 301 car data 246, 251, 275 Chebyshev polynomial 178 curb weight 246 diesel fuel 251 diffusion limited aggregation (DLA) 6 draft lottery 248 European Union 262 flow balance 287 fractal 6, 171 fuel consumption 275 glass data 265 ice cream 111 lottery 248 353 example data sets (continued) mammals 254 Mandelbrot set 171 marathon 4, 280 miles per gallon (mpg) 275 price 246 spectrum 104 sunspots 284 web traffic 252 examples axes 123 enhanced text mode 204 fit command 195 scatter plots 246, 248 executing commands in subshells 225 exit command 309 exp kernel 150 explanations 102 data files 102 plot command 102 explicit mode 158 exploratory data analysis (EDA) 11 exponential functions 316 export script 27 exporting 25, 201 expressions (math) 38 F false-color plot 161 features abbreviations 23 autoscale 94 multiplot 175 sensible defaults 23 smooth cumulative 260 field separators 52 fig terminal 220 file formats EPS 210 GIF 207 JPG 208 PDF 217 PNG 207 PostScript 210 SVG 208 file system (commands) 310 files data 51 data sets 30 examples of data 20 exporting graphs to 201 initialization 235 input data 296 loading 25 output 298 plotting data from 20 plotting unsorted data 32 reading palettes from 156 reading tic labels from 122 saving 25 filled plot styles 81 financebars style 80 fit command 191, 314 control variables 193 environmental variables 194 example 195 options 194 output variables 193 tips 194 fit option 322 flow balance 287 flt type indicator xxviii flushing output channels 205 font default 202 directive 206 PostScript 202, 210–211, 216 specifying 210 terminals 202 TrueType 202, 206–207, 211 font tables 232, 341 fontpath option 207, 322 forcing interactive sessions 229 format cb option 159 format option 331 formats data files 51 grid 146 matrix 148 formatting tic labels 118 fractal 6, 171 freefont project 207 frequency directive 33, 35, 257 fsteps style 72 fuel consumption 275 functions Bessel 317 built-in 38 column 42 column manipulation 319 complex arguments 318 creating palettes with 155 cumulative distribution 259 error integral 317 exponential 316 gamma 317 gprint() 122 hyperbolic 316 imag() 41 keyword 155 logarithmic 316 miscellaneous 318 plotting 43 rand() 38, 256 real() 41 scanf() 128 smooth frequency 256 smooth kdensity 258 strftime() 127 strings 56, 318 system() 225 time column handling 319 trigonometric 316 user-defined 39 valid() 43 xticlabels() 122 yticlabels() 122, 263 G gamma functions 317 Gaussian kernel 150, 258 generating logarithmic plots 44 textual output 59 GIF terminal 207 glass data 265 global plot styles 68, 70 GNU software, compared with gnuplot 13 gnuplot 3, 13 benefits 14 building 303 calling 228 as CGI script 239 command history 61 compared with GNU software 13 configuring 307 examples 4, 6 help 61 hot keys 62 installing 303 invoking 17, 229 limitations 15 mailing list 14, 345 mousing 62–63 new features 14 obtaining 303 tricks and warnings 44 web pages 239 web sites 305 354 gpic terminal 220 gprint() function 122 Grace graphing tool 349 graphical analysis 3, 9–10 benefits 12 limitations 12 resources 345 graphical analysis techniques 273 banking 284 changing appearance 284 changing compositions problems 292 comparing data 278 core principle 274 enhancing quantitative perception 289 housekeeping 296 iteration 275 judging lengths/ distances 287 plot ranges 291 presentation graphics 298 responsiveness 280 transformation 275 truncation 280 zero in plot range 291 graphical methods 245 counting statistics 256 multivariate data 264 ranked data 262 relationships 246 graphics file formats 206 presentation 11, 298 graphicx package (LaTeX) 212 graphing tools 348 graphs 92 aligned on common axis 181 arrays with layout 177 arrows 94 components 92 coordinates 93 creating with palettes 157 decorations 94 exporting 25 exporting to file 201 key 100 legend 100 lifecycle 296 objects 99 pm3d mode 157 polar mode 185 presentation 10 scatter plot 23 stacked 293 text labels 97 understanding data with 10, 301 within graphs 179 grid axes 123 grid cbtics option 159 grid format 146 grid mcbtics option 159 grid option 332 H half-tone shading example 87 hann kernel 150 hardware requirements xxix head option 95 header option 216 heat scale palette 165 help command 61, 309 hidden3d option 136, 335 histeps style 72 histograms 74, 256 history command 61, 309 history feature 17 historysize option 322 hot key bindings, creating custom 236 hot keys 62 housekeeping 296 graph lifecycle 296 input data files 296 output files 298 hue-based palette 166 hue-saturation-value (HSV) scheme 153, 165 hyperbolic functions 316 I ice cream 111 idx type indicator xxviii if command 315 imag() function 41 image analysis 11 implicit mode 158 impulses style 73 including EPS files in LaTeX documents 211 index directive 31 indexing strings 56 initialization file 235 inline plot styles 68 input axes 128 data files 296 redirection 226 insets 176 INSTALL file 306 INSTALL.gnu file 306 installing 308 gnuplot 303 on Linux 304 on Mac OS X 304 on Windows 304 int type indicator xxviii interactive terminals 218 options 343 interlacing 207 interpolate keyword 158 interpolating between colors 154 interpolation curves 37 invoking gnuplot 17, 229 isosamples option 136, 335 iteration 273, 275 case study 275 defined 10 J jitter plots 256 JPG terminal 208 judging lengths/distances 287 K kdensity directive 258 Kelley, Colin (software developer) 13 kernel 258 density estimates 258 Gaussian 258 smoothing 150 key 22, 100 appearance 104 default settings 104 explanation 102 layout 101 option 327 placement 101 turning on/off 101 keyboard event 238 keywords butt 205 columnsfirst 177 corners2color 158 dashed 205 default 159 downwards 177 dynamic 209 functions 155 355 keywords (continued) interpolate 158 offset 138, 178 rounded 205 rowsfirst 177 scale 178 solid 205 title 22 trianglepattern 138 upwards 177 knots, splines 249 kst graphing tool 348 L _label option 332 label option 327 labels (scatter plots) 251 landscape option 209 LaTeX EPS file 211 PostScript plots 211 tricks 217 layout directive 177 key 101 source tree 306 least squares fitting 191 legend.

In general, the colors are distributed rather uniformly over the entire spectrum, because this matches up with the regularly varying function in this plot. 9.4.2 A complex figure As an example of a graph that includes a lot of fine detail, I’ve chosen a section from the edge of the Mandelbrot set. The Mandelbrot set is the set of all points in the complex plane for which a certain simple iteration process stays bounded. What’s noteworthy here is that the border between points inside the set and outside of it isn’t smooth—in fact the border is “infinitely” complicated, showing details at all levels of magnification.6 For points far from the Mandelbrot set, the iteration will diverge quickly (after just a few steps). But as we approach the border, the iteration will take many more steps before finally diverging. Once inside the set, the iteration doesn’t diverge (that’s the definition of the Mandelbrot set, after all). The input to the data visualization project we want to study in this section is a file containing the x and y coordinates of all points in a certain region of the complex plane, together with the number of steps that were required before the iteration diverged.

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


accounting loophole / creative accounting, banking crisis, banks create money, barriers to entry, Benoit Mandelbrot, Big bang: deregulation of the City of London, Black Swan, Bonfire of the Vanities, butterfly effect, capital asset pricing model, cellular automata, central bank independence, citizen journalism, clockwork universe, collective bargaining, complexity theory, correlation coefficient, creative destruction, credit crunch, David Ricardo: comparative advantage, debt deflation, diversification, double entry bookkeeping,, Eugene Fama: efficient market hypothesis, experimental subject, Financial Instability Hypothesis, fixed income, Fractional reserve banking, full employment, Henri Poincaré, housing crisis, Hyman Minsky, income inequality, information asymmetry, invisible hand, iterative process, John von Neumann, laissez-faire capitalism, liquidity trap, Long Term Capital Management, mandelbrot fractal, margin call, market bubble, market clearing, market microstructure, means of production, minimum wage unemployment, money market fund, open economy, Pareto efficiency, Paul Samuelson, place-making, Ponzi scheme, profit maximization, quantitative easing, RAND corporation, random walk, risk tolerance, risk/return, Robert Shiller, Robert Shiller, Ronald Coase, Schrödinger's Cat, scientific mainstream, seigniorage, six sigma, South Sea Bubble, stochastic process, The Great Moderation, The Wealth of Nations by Adam Smith, Thorstein Veblen, time value of money, total factor productivity, tulip mania, wage slave, zero-sum game

., Jr (2004) ‘Keynote address to the 2003 HOPE Conference: my Keynesian education,’ History of Political Economy, 36: 12–24. Mackay, C. (1841) Extraordinary Popular Delusions and the Madness of Crowds, New York: Crown Trade Paperbacks, Mandel, E. (1971) The Formation of the Economic Thought of Karl Marx, London: NLB. Mandelbrot, B. (1971) ‘Linear regression with non-normal error terms: a comment,’ Review of Economics and Statistics, 53(2): 205–6. Mandelbrot, B. (2005) ‘The inescapable need for fractal tools in finance,’ Annals of Finance, 1(2): 193–5. Mandelbrot, B. B. and R. L. Hudson (2004) The (Mis)behaviour of Markets: A fractal view of risk, ruin and reward, London: Profile. Mankiw, N. G. (2008) Principles of Microeconomics, 5E, Boston, MA: South-Western College Publishers. Mantel, R. R. (1974) ‘On the characterisation of aggregate excess demand,’ Journal of Economic Theory, 7: 348–53.

Economics does not generate a sufficient volume of data, but financial markets do in abundance, with the price and volume data of financial transactions; as Joe McCauley put it, ‘the concentration is on financial markets because that is where one finds the very best data for a careful empirical analysis’ (McCauley 2004: xi). Given that it is a relatively new field, there are numerous explanations of the volatility of financial markets within Econophysics – including Power Law models of stock market movements (Gabaix, Gopikrishnan et al. 2006), Didier Sornette’s earthquake-based analysis (Sornette 2003), Joe McCauley’s empirically derived Fokker-Planck model (McCauley 2004), and Mandelbrot’s fractal geometry (Mandelbrot and Hudson 2004) – and it would require another book to detail them all. A unifying theme is that the behavior of financial markets is driven by the interactions of numerous market participants with each other, and these generate a highly unstable and therefore relatively unpredictable time series in financial data themselves. These characteristics resemble the behavior of fissile materials in a nuclear reactor, or tectonic plates in an earthquake zone, physical processes for which physicists have developed an enormous arsenal of mathematical analytic techniques in the last century.

This Econophysics explanation of the unpredictability of finance markets is thus diametrically opposed to the explanation that neoclassical economics has given of precisely the same phenomenon – the difficulty of predicting the market – and Econophysicists react with incredulity to the simplistic ‘random disturbances to an equilibrium process’ explanation that neoclassical economists provide: Three states of matter – solid, liquid, and gas – have long been known. An analogous distinction between three states of randomness – mild, slow and wild – arises from the mathematics of fractal geometry. Conventional finance theory assumes that variation of prices can be modeled by random processes that, in effect, follow the simplest ‘mild’ pattern, as if each uptick or downtick were determined by the toss of a coin. What fractals show […] is that by that standard, real prices ‘misbehave’ very badly. A more accurate, multifractal model of wild price variation paves the way for a new, more reliable type of financial theory. (Mandelbrot and Hudson 2004: v) Economists teach that markets can be described by equilibrium. Econophysicists teach that markets are very far from equilibrium and are dynamically complex […] equilibrium is never a good approximation […] market equilibrium does not and cannot occur […] (McCauley 2004: 185) Uncertainties and variabilities are the key words to describe the ever-changing environments around us.

The Singularity Is Near: When Humans Transcend Biology by Ray Kurzweil


additive manufacturing, AI winter, Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, anthropic principle, Any sufficiently advanced technology is indistinguishable from magic, artificial general intelligence, Asilomar, augmented reality, autonomous vehicles, Benoit Mandelbrot, Bill Joy: nanobots, bioinformatics, brain emulation, Brewster Kahle, Brownian motion, business intelligence,, call centre, carbon-based life, cellular automata, Claude Shannon: information theory, complexity theory, conceptual framework, Conway's Game of Life, cosmological constant, cosmological principle, cuban missile crisis, data acquisition, Dava Sobel, David Brooks, Dean Kamen, disintermediation, double helix, Douglas Hofstadter,, epigenetics, factory automation, friendly AI, George Gilder, Gödel, Escher, Bach, informal economy, information retrieval, invention of the telephone, invention of the telescope, invention of writing, Isaac Newton, iterative process, Jaron Lanier, Jeff Bezos, job automation, job satisfaction, John von Neumann, Kevin Kelly, Law of Accelerating Returns, life extension, lifelogging, linked data, Loebner Prize, Louis Pasteur, mandelbrot fractal, Mikhail Gorbachev, mouse model, Murray Gell-Mann, mutually assured destruction, natural language processing, Network effects, new economy, Norbert Wiener, oil shale / tar sands, optical character recognition, pattern recognition, phenotype, premature optimization, randomized controlled trial, Ray Kurzweil, remote working, reversible computing, Richard Feynman, Richard Feynman, Robert Metcalfe, Rodney Brooks, Search for Extraterrestrial Intelligence, selection bias, semantic web, Silicon Valley, Singularitarianism, speech recognition, statistical model, stem cell, Stephen Hawking, Stewart Brand, strong AI, superintelligent machines, technological singularity, Ted Kaczynski, telepresence, The Coming Technological Singularity, Thomas Bayes, transaction costs, Turing machine, Turing test, Vernor Vinge, Y2K, Yogi Berra

The primary problem with Bell's perspective is that he fails to account for the self-organizing, chaotic, and fractal nature of the brain's design. It's certainly true that the brain is complex, but a lot of the complication is more apparent than real. In other words, the principles of the design of the brain are simpler than they appear. 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.

I would agree that the roughly thirty to one hundred million bytes of information in the genome do not represent a simple design (certainly far more complex than the six characters in the definition of the Mandelbrot set), but it is a level of complexity that we can already manage with our technology. Many observers are confused by the apparent complexity in the brain's physical instantiation, failing to recognize that the fractal nature of the design means that the actual design information is far simpler than what we see in the brain. I also mentioned in chapter 2 that the design information in the genome is a probabilistic fractal, meaning that the rules are applied with a certain amount of randomness each time a rule is iterated. There is, for example, very little information in the genome describing the wiring pattern for the cerebellum, which comprises more than half the neurons in the brain.

This process is repeated many times, with each newly created element of a generator becoming an initiator and being replaced with a new scaled generator. Each new generation of fractal expansion adds apparent complexity but requires no additional design information. A probabilistic fractal adds the element of uncertainty. Whereas a deterministic fractal will look the same every time it is rendered, a probabilistic fractal will look different each time, although with similar characteristics. In a probabilistic fractal, the probability of each generator element being applied is less than 1. In this way, the resulting designs have a more organic appearance. Probabilistic fractals are used in graphics programs to generate realistic-looking images of mountains, clouds, seashores, foliage, and other organic scenes. A key aspect of a probabilistic fractal is that it enables the generation of a great deal of apparent complexity, including extensive varying detail, from a relatively small amount of design information.

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


Albert Einstein, Benoit Mandelbrot, correlation does not imply causation, discovery of DNA, double helix, Drosophila, epigenetics, Isaac Newton, Mahatma Gandhi, mandelbrot fractal, Mars Rover, On the Revolutions of the Heavenly Spheres, phenotype, placebo effect, randomized controlled trial, selective serotonin reuptake inhibitor (SSRI), stem cell

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 same equation is then repeated ad infinitum. For example, the “Mandelbrot set” is based on the simple formula of taking a number, multiplying it by itself and then adding the original number. The result of that equation is then used as the input of the subsequent equation; the result of that equation is then used as the input for the next equation and so on.

The challenge is that even though each equation follows the same formula, these equations must be repeated millions of times to actually visualize a fractal pattern. The manual labor and time needed to complete millions of equations prevented early mathematicians from recognizing the value of fractal geometry. With the advent of powerful computers Mandelbrot was able to define this new math. Inherent in the geometry of fractals is the creation of ever-repeating, “self-similar” patterns nested within one another. You can get a rough idea of the repeating shapes by picturing the eternally popular toy, hand-painted Russian nesting dolls. Each smaller structure is a miniature, but not necessarily an exact version of the larger form. Fractal geometry emphasizes the relationship between the patterns in a whole structure and the patterns seen in parts of a structure.

Evolution, the expansion of awareness, can then be physically defined by the increase of membrane surface area. Mathematical studies have found that fractal geometry is the best way to get the most surface area (membrane) within a three-dimensional space (cell). Therefore, evolution becomes a fractal affair. Repeating patterns in nature are a necessity, not a coincidence, of “fractal” evolution. My point is not to get caught up in the mathematical details of the modeling. There are repetitive fractal patterns in nature and in evolution as well. The strikingly beautiful, computer-generated pictures that illustrate fractal patterns should remind us that, despite our modern angst and the seeming chaos of our world, there is order in nature, and there is nothing truly new under the sun. Evolution’s repetitive, fractal patterns allow us to predict that humans will figure out how to expand their consciousness in order to climb another rung of the evolutionary ladder.

pages: 484 words: 136,735

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


bank run, banking crisis, Benoit Mandelbrot, Berlin Wall, Black Swan, bonus culture, Bretton Woods, BRICs, Carmen Reinhart, cognitive dissonance, collapse of Lehman Brothers, Corn Laws, correlation does not imply causation, creative destruction, credit crunch, currency manipulation / currency intervention, David Ricardo: comparative advantage, deglobalization, Deng Xiaoping, Edward Glaeser, Eugene Fama: efficient market hypothesis, eurozone crisis, experimental economics, F. W. de Klerk, failed state, Fall of the Berlin Wall, financial deregulation, financial innovation, Financial Instability Hypothesis, floating exchange rates, full employment, George Akerlof, global rebalancing, Hyman Minsky, income inequality, information asymmetry, invisible hand, Isaac Newton, Joseph Schumpeter, Kenneth Arrow, Kenneth Rogoff, laissez-faire capitalism, Long Term Capital Management, mandelbrot fractal, market design, market fundamentalism, Martin Wolf, money market fund, moral hazard, mortgage debt, new economy, Northern Rock, offshore financial centre, oil shock, paradox of thrift, Pareto efficiency, Paul Samuelson, peak oil,, Ponzi scheme, post-industrial society, price stability, profit maximization, profit motive, quantitative easing, Ralph Waldo Emerson, random walk, rent-seeking, reserve currency, rising living standards, Robert Shiller, Robert Shiller, Ronald Reagan, shareholder value, short selling, South Sea Bubble, sovereign wealth fund, special drawing rights, statistical model, The Chicago School, The Great Moderation, The inhabitant of London could order by telephone, sipping his morning tea in bed, the various products of the whole earth, The Wealth of Nations by Adam Smith, Thomas Kuhn: the structure of scientific revolutions, too big to fail, Vilfredo Pareto, Washington Consensus, zero-sum game

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. Mandelbrot, in his book The Misbehaviour of Markets, described how forty years of effort to interest economists in fractal geometry were ridiculed or ignored, despite the fact that they seemed to provide a much better analysis of extreme market behavior than standard methods.

Rather they are ‘fixed’—amended, qualified, particularized, expanded and complicated. Bit by bit, from a bad seed a big but sickly tree is built with glue, nails, screws and scaffolding. Conventional economics assumes the financial system is a linear, continuous, rational machine and these false assumptions are built into the risk models used by many of the world’s banks.1 Despite the success achieved by fractal geometry and nonlinear modeling in the study of earthquakes, weather, evolution, ecology, and other complex systems, Mandelbrot always faced the same objection from economists when he proposed applying similar techniques to markets. These non-Gaussian mathematical methods could only provide approximations, as opposed to the precise answers offered by the Efficient Market Hypothesis and Gaussian statistics.2 The fact that the exact answers of EMH bore no relation to reality did not seem to deter “scientific” economists.

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 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. Sawyer, The Economics of Michal Kalecki.

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


23andMe, Albert Einstein, Alfred Russel Wallace, banking crisis, Barry Marshall: ulcers, Benoit Mandelbrot, Berlin Wall, biofilm, Black Swan, butterfly effect, Cass Sunstein, cloud computing, congestion charging, correlation does not imply causation, Daniel Kahneman / Amos Tversky, dark matter, data acquisition, David Brooks, delayed gratification, Emanuel Derman, epigenetics, Exxon Valdez, Flash crash, Flynn Effect, hive mind, impulse control, information retrieval, Intergovernmental Panel on Climate Change (IPCC), Isaac Newton, Jaron Lanier, John von Neumann, Kevin Kelly, lifelogging, mandelbrot fractal, market design, Mars Rover, Marshall McLuhan, microbiome, Murray Gell-Mann, Nicholas Carr, open economy, Pierre-Simon Laplace, place-making, placebo effect, pre–internet, QWERTY keyboard, random walk, randomized controlled trial, rent control, Richard Feynman, Richard Feynman, Richard Feynman: Challenger O-ring, Richard Thaler, Satyajit Das, Schrödinger's Cat, security theater, selection bias, Silicon Valley, stem cell, Steve Jobs, Steven Pinker, Stewart Brand, the scientific method, Thorstein Veblen, Turing complete, Turing machine, Vilfredo Pareto, Walter Mischel, Whole Earth Catalog, zero-sum game

There was unexpected complexity in a single note struck on a piano—complexity that eluded even dozens of oscillators and filters by magnitudes. 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.”

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. If you draw lines on a map to approximate the coastline, you get one length, but if you try to measure the real bumps in every inlet and bay, the curves of each pebble and grain of sand, you get a much different length.

Remarkably, once the dust began to settle, it became apparent that the statistical properties of the resulting Internet were quite special: The time delays for packet transmission, the network topology, and even the information transmitted exhibit fractal properties. However you look at the Internet—locally or globally, on short time scales or long—it looks exactly the same. Although the discovery of this fractal structure, around 1995, was an unwelcome surprise because standard traffic-control algorithms, as used by routers, were designed assuming that all properties of the network dynamics would be random, the fractality is also broadly characteristic of biological networks. Without a master blueprint, the evolution of an Internet is subject to the same underlying statistical laws that govern biological evolution, and structure emerges spontaneously, without the need for a controlling entity.

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The Simpsons and Their Mathematical Secrets by Simon Singh


Albert Einstein, Andrew Wiles, Benoit Mandelbrot, cognitive dissonance, Donald Knuth, Erdős number, Georg Cantor, Grace Hopper, Isaac Newton, John Nash: game theory, mandelbrot fractal, Menlo Park, Norbert Wiener, Norman Mailer, P = NP, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, Richard Feynman, Richard Feynman, Rubik’s Cube, Schrödinger's Cat, Simon Singh, Stephen Hawking, Wolfskehl Prize, women in the workforce

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.” Fractals are also recognizable because they exhibit fractional dimensions. To get a sense of what it means to have fractional dimensionality, we will examine a particular fractal object, namely the Sierpinski triangle, which is constructed according to the following recipe.

They can use the dimensional drift to get out of the second dimension and into the third dimension. We actually have this amazing sequence, because they fly through this huge fractal landscape that represents the area between two dimensions and three dimensions. The scene contains some pretty amazing computer graphics.” The fractal landscape is particularly appropriate, because fractals actually exhibit a fractional dimensionality. The fractal landscape appears on the journey between the two-dimensional and three-dimensional worlds, which is exactly where one might expect to find a fractional dimension. If you want to know more about fractals, please refer to appendix 4, where there is a very brief overview of this topic, focusing particularly on how an object can possibly be fractionally dimensional. The Möbius strip in “2-D Blacktop” resonates with a mathematical concept that appears in “The Route of All Evil” (2002).

Joke 5 Q: What’s purple, dangerous, and commutes? 1 point A: An abelian grape with a machine gun. Joke 6 Q: What’s big, grey, and proves the uncountability of the decimal numbers? 2 points A: Cantor’s diagonal elephant. Joke 7 Q: What’s the world’s longest song? 2 points A: “0 Bottles of Beer on the Wall.” Joke 8 Q: What does the “B.” in Benoit B. Mandelbrot stand for? 4 points A: Benoit B. Mandelbrot. Joke 9 Q: What do you call a young eigensheep? 1 point A: A lamb, duh! Joke 10 One day, ye director of ye royal chain mail factory was asked to submit a sample in order to try to win a very large order for chain mail tunics and leggings. Though the tunic sample was accepted, he was told that the leggings were too long. He submitted a new sample, and this time the leggings were better, but too short.

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Power, Sex, Suicide: Mitochondria and the Meaning of Life by Nick Lane


Benoit Mandelbrot, clockwork universe, double helix, Drosophila, Geoffrey West, Santa Fe Institute, Louis Pasteur, mandelbrot fractal, out of africa, phenotype, random walk, Richard Feynman, Richard Feynman, stem cell, unbiased observer

The usual answer is that within species the metabolic rate does indeed vary with 2/3, and the 3/4 exponent only becomes apparent when we compare different species. 1 The Power Laws of Biology 161 Laboratory, Geoffrey West, joined forces with the ecologists James Brown and Brian Enquist, at the University of New Mexico, Albuquerque (through the Santa Fe Institute, an organization that fosters cross-disciplinary collaborations). They came up with a radical explanation based on the fractal geometry of branching supply networks, such as the circulatory system of mammals, the respiratory tubes of insects (the trachea), and the plant vascular system. 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.

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? Fractals can also be generated mathematically, often by using a reiterative geometric formula to specify the angle and density of branches (the ‘fractal dimension’). Both types of fractal share a property known as scale invariance, which is to say they ‘look’ similar whatever the magnification. For example, the contours of a rock often resemble those of a cliff or even a mountain, and for this reason geologists like to leave a hammer lying around in photographs, to enable viewers to grasp the scale.

Gould, Stephen Jay. Full House. Random House, New York, USA, 1997. Haldane, J. B. S. On Being the Right Size, ed. John Maynard-Smith. Oxford University Press, Oxford, UK, 1985. Mandelbrot, Benoit. The Fractal Geometry of Nature. W. H. Freeman, New York, 1977. Ridley, Mark. Mendel’s Demon. Weidenfeld & Nicolson, London, UK, 2000. The power laws of biology Bennett, A. F. Structural and functional determinates of metabolic rate. American Zoologist 28: 699–708; 1988. 336 Further Reading Heusner, A. Size and power in mammals. Journal of Experimental Biology 160: 25–54; 1991. Kleiber, M. The Fire of Life. Wiley, New York, USA, 1961. Fractal geometry and scaling Banavar, J., Damuth, J., Maritan, A., and Rinaldo, A. Supply-demand balance and metabolic scaling. Proceedings of the National Academy of Sciences USA 99: 10506–10509; 2002.

pages: 518 words: 107,836

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


Albert Einstein, Andrei Shleifer, Benoit Mandelbrot, bitcoin, Brownian motion, Claude Shannon: information theory, cloud computing, cognitive dissonance, computer age, conceptual framework, continuation of politics by other means, crony capitalism, crowdsourcing, cuban missile crisis, Daniel Kahneman / Amos Tversky, David Graeber, Dissolution of the Soviet Union, Donald Davies, double helix, Drosophila, Francis Fukuyama: the end of history, From Mathematics to the Technologies of Life and Death, hive mind, index card, informal economy, information asymmetry, invisible hand, Jacquard loom, Jacquard loom, John von Neumann, Kevin Kelly, knowledge economy, knowledge worker, linear programming, mandelbrot fractal, Marshall McLuhan, means of production, Menlo Park, Mikhail Gorbachev, mutually assured destruction, Network effects, Norbert Wiener, packet switching, Pareto efficiency, pattern recognition, Paul Erdős, Peter Thiel, Philip Mirowski, RAND corporation, rent-seeking, road to serfdom, Ronald Coase, scientific mainstream, Steve Jobs, Stewart Brand, stochastic process, technoutopianism, The Structural Transformation of the Public Sphere, transaction costs, Turing machine

The imprint of cybernetics can still be seen in subsequent generations of French theorists. These postwar happenings are described briefly below. 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.)

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. And finally, Wiener’s own domineering and brilliant father, Leo Wiener, was a self-made polymath, the preeminent translator of Tolstoy into English in the twentieth-century, the founder of Slavic studies in America, an émigré from a Belarusian shtetl, and like his son, a humanist committed to uncovering methods for nearly universal communication.41 Although summarizing the intellectual and international sources for the consolidation of cybernetics as a midcentury science for self-governing systems is beyond the scope of this project, the following statement is probably not too far of a stretch.

., 19, 27, 91, 97 Liebniz, Gottlob, 67 Linear modeling, 68–69, 205 Linear programming, 68 Llull, Ramon, 42 “A Logical Calculus of the Ideas Immanent in Nervous Activity,” 18 Losev, V., 119 L’viv System, 154 Lyapunov, Aleksei, 34–36, 40, 43–44, 46, 82, 85, 89, 169, 183, 216 Lysenko, Trofim, 32, 40 MacKay, Donald, 27 Macropiping, 118 Macy Conferences on Cybernetics, 18–19 “The Main Features of Cybernetics,” 35–39 Malinovksy, Boris, 117, 154, 166–167 “Man-Computer Symbiosis,” 91 Mandelbrot, Benoit, 25, 28 Mandelbrot, Szolem, 25, 28 Mansfield amendments, 93 Market economy, 22 “Mark III, a Calculator,” 30 Markov, Andrei, Jr., 34, 42, 46 Marx, Karl, 58, 65, 74, 199, 204 Marxism-Leninism, 33, 139, 194–195 Mar’yanovich, T. P., 118 Materialism, 39–40 “A Mathematical Theory of Communication,” 98 Matiukhin, Nikolai, 84–85, 91 “The Matter of the Whole Country,” 167 Maturana, Humberto, 27, 96 McCarthy, John, 178 McCulloch, Warren, 18–19, 22–23, 27, 54, 95–96, 100, 119–120, 193, 202 Mead, Margaret, 19 Media technologies, 205–206 Mendeleev, Dmitri, 32 Merton, Robert, 97 MESM (malaya electronicheskaya schetnaya mashina), 126, 128 Messages, theory of, 17–18, 20 Microcomputers, 127.

pages: 372 words: 101,174

How to Create a Mind: The Secret of Human Thought Revealed by Ray Kurzweil


Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, Albert Michelson, anesthesia awareness, anthropic principle, brain emulation, cellular automata, Claude Shannon: information theory, cloud computing, computer age, Dean Kamen, discovery of DNA, double helix,, epigenetics, George Gilder, Google Earth, Isaac Newton, iterative process, Jacquard loom, Jacquard loom, John von Neumann, Law of Accelerating Returns, linear programming, Loebner Prize, mandelbrot fractal, Norbert Wiener, optical character recognition, pattern recognition, Peter Thiel, Ralph Waldo Emerson, random walk, Ray Kurzweil, reversible computing, selective serotonin reuptake inhibitor (SSRI), self-driving car, speech recognition, Steven Pinker, strong AI, the scientific method, theory of mind, Turing complete, Turing machine, Turing test, Wall-E, Watson beat the top human players on Jeopardy!, X Prize

For a more complete description of this argument, see the section “[The Impact…] on the Intelligent Destiny of the Cosmos: Why We Are Probably Alone in the Universe” in chapter 6 of The Singularity Is Near by Ray Kurzweil (New York: Viking, 2005). 5. James D. Watson, Discovering the Brain (Washington, DC: National Academies Press, 1992). 6. Sebastian Seung, Connectome: How the Brain’s Wiring Makes Us Who We Are (New York: Houghton Mifflin Harcourt, 2012). 7. “Mandelbrot Zoom,”; “Fractal Zoom Mandelbrot Corner,” Chapter 1: Thought Experiments on the World 1. Charles Darwin, The Origin of Species (P. F. Collier & Son, 1909), 185/95–96. 2. Darwin, On the Origin of Species, 751 (206.1.1-6), Peckham’s Variorum edition, edited by Morse Peckham, The Origin of Species by Charles Darwin: A Variorum Text (Philadelphia: University of Pennsylvania Press, 1959). 3.

Consider the famous Mandelbrot set, the image of which has long been a symbol of complexity. To appreciate its apparent complication, it is useful to zoom in on its image (which you can access via the links in this endnote).7 There is endless intricacy within intricacy, and they are always different. Yet the design—the formula—for the Mandelbrot set couldn’t be simpler. It is six characters long: Z = Z2 + C, in which Z is a “complex” number (meaning a pair of numbers) and C is a constant. It is not necessary to fully understand the Mandelbrot function to see that it is simple. This formula is applied iteratively and at every level of a hierarchy. The same is true of the brain. Its repeating structure is not as simple as that of the six-character formula of the Mandelbrot set, but it is not nearly as complex as the millions of quotations on the brain’s complexity would suggest.

Its repeating structure is not as simple as that of the six-character formula of the Mandelbrot set, but it is not nearly as complex as the millions of quotations on the brain’s complexity would suggest. This neocortical design is repeated over and over at every level of the conceptual hierarchy represented by the neocortex. Einstein articulated my goals in this book well when he said that “any intelligent fool can make things bigger and more complex…but it takes…a lot of courage to move in the opposite direction.” One view of the display of the Mandelbrot set, a simple formula that is iteratively applied. As one zooms in on the display, the images constantly change in apparently complex ways. So far I have been talking about the brain. But what about the mind? For example, how does a problem-solving neocortex attain consciousness? And while we’re on the subject, just how many conscious minds do we have in our brain? There is evidence that suggests there may be more than one.

pages: 374 words: 114,600

The Quants by Scott Patterson


Albert Einstein, asset allocation, automated trading system, beat the dealer, Benoit Mandelbrot, Bernie Madoff, Bernie Sanders, Black Swan, Black-Scholes formula, Bonfire of the Vanities, Brownian motion, buttonwood tree, buy low sell high, capital asset pricing model, centralized clearinghouse, Claude Shannon: information theory, cloud computing, collapse of Lehman Brothers, collateralized debt obligation, commoditize, computerized trading, Credit Default Swap, credit default swaps / collateralized debt obligations, diversification, Donald Trump, Doomsday Clock, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, fixed income, Gordon Gekko, greed is good, Haight Ashbury, I will remember that I didn’t make the world, and it doesn’t satisfy my equations, index fund, invention of the telegraph, invisible hand, Isaac Newton, job automation, John Meriwether, John Nash: game theory, law of one price, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, merger arbitrage, money market fund, Myron Scholes, NetJets, new economy, offshore financial centre, old-boy network, Paul Lévy, Paul Samuelson, Ponzi scheme, quantitative hedge fund, quantitative trading / quantitative finance, race to the bottom, random walk, Renaissance Technologies, risk-adjusted returns, Rod Stewart played at Stephen Schwarzman birthday party, Ronald Reagan, Sergey Aleynikov, short selling, South Sea Bubble, speech recognition, statistical arbitrage, The Chicago School, The Great Moderation, The Predators' Ball, too big to fail, transaction costs, value at risk, volatility smile, yield curve, éminence grise

The fact was that they existed and cropped up time and again in all sorts of markets. The upshot of Mandelbrot’s research was that markets are far less well behaved than standard financial theory held. Out at the no-man’s-land on the wings of the bell curve lurked a dark side of markets that haunted the quants like a bad dream, one many had seemingly banished into subconsciousness. Mandelbrot’s message had been picked up years later by Nassim Taleb, who repeatedly warned quants that their models were doomed to fail because unforeseen black swans (which reputedly didn’t exist) would swoop in from nowhere and scramble the system. Such notions threatened to devastate the elegant mathematical world of quants such as Cootner and Fama. Mandelbrot had been swiftly attacked, and—though he remained a mathematical legend and created an entire new field known as fractal geometry and pioneering discoveries in the science of chaos—was soon forgotten in the world of quants as little more than a footnote in their long march to victory.

Another source is “The Evolution of Portfolio Insurance,” by Hayne E. Leland and Mark Rubinstein, published in Portfolio Insurance: A Guide to Dynamic Hedging, edited by Donald Luskin (John Wiley & Sons, 1988). “Even if one were to have lived”: The age of the universe is 13.5 billion years, not 20 billion. When German tanks rumbled into France: Some details of Mandelbrot’s life come from a series of interviews with Mandelbrot in the summer of 2008. Many also come from the book The (Mis)Behavior of Markets: A Fractal View of Financial Turbulence, by Benoit Mandelbrot and Richard L. Hudson (Basic Books, 2006). “I realized that the existence of the smile”: My Life as a Quant, by Emanuel Derman (John Wiley & Sons, 2004), 226. A squad of fifty armed federal marshals: Certain details come from Den of Thieves, by James Stewart (Simon & Schuster, 1991). He also worked as a consultant: I learned the fascinating story of Thorp’s discovery of the Madoff fraud in several interviews with Thorp in December 2008 as the fraud was discovered.

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. Young Mandelbrot studied under his uncle and entered a French secondary school. But his life was upended when the Nazis invaded. As the Germans closed in, the Mandelbrot family fled to the small hill town of Tulle in southwest France, where they had friends. Benoit enrolled in the local school, where there was little competition.

pages: 695 words: 194,693

Money Changes Everything: How Finance Made Civilization Possible by William N. Goetzmann


Albert Einstein, Andrei Shleifer, asset allocation, asset-backed security, banking crisis, Benoit Mandelbrot, Black Swan, Black-Scholes formula, Bretton Woods, Brownian motion, capital asset pricing model, Cass Sunstein, collective bargaining, colonial exploitation, compound rate of return, conceptual framework, corporate governance, Credit Default Swap, David Ricardo: comparative advantage, debt deflation, delayed gratification, Detroit bankruptcy, disintermediation, diversified portfolio, double entry bookkeeping, Edmond Halley,, equity premium, financial independence, financial innovation, financial intermediation, fixed income, frictionless, frictionless market, full employment, high net worth, income inequality, index fund, invention of the steam engine, invention of writing, invisible hand, James Watt: steam engine, joint-stock company, joint-stock limited liability company, laissez-faire capitalism, Louis Bachelier, mandelbrot fractal, market bubble, means of production, money market fund, money: store of value / unit of account / medium of exchange, moral hazard, Myron Scholes, new economy, passive investing, Paul Lévy, Ponzi scheme, price stability, principal–agent problem, profit maximization, profit motive, quantitative trading / quantitative finance, random walk, Richard Thaler, Robert Shiller, Robert Shiller, shareholder value, short selling, South Sea Bubble, sovereign wealth fund, spice trade, stochastic process, the scientific method, The Wealth of Nations by Adam Smith, Thomas Malthus, time value of money, too big to fail, trade liberalization, trade route, transatlantic slave trade, transatlantic slave trade, tulip mania, wage slave

It was a nice refinement of Regnault’s hypothesis articulated almost precisely a century prior. Although Mandelbrot ultimately developed a fractal-based option-pricing model with two of his students that allowed for extreme events and a more general stochastic process, for various reasons Mandelbrot never saw it adopted in practice to any great extent. I suspect that this is because the solution, while potentially useful, is complicated and contradicts most other tools that quantitative financiers use. With Mandelbrot’s models, it is all or nothing. You have to take a leap beyond the world of Brownian motion and throw out old friends like Bernoulli’s law of large numbers. For most quants in practice (and professors studying the markets), the leap is too great, and the payoff in terms of understanding may not be sufficient. After all, Mandelbrot never promised that the timing of a giant crash could be predicted, just that it would likely happen.

A drop of 22% in a couple of days was not in Mark Rubenstein’s game plan, because standard models used for option pricing effectively assume that the logarithm of stock prices are “normal”—that is, they conform to the standard bell-curve distribution. 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. Lévy’s research focused on “stochastic processes”: mathematical models that describe the behavior of some variable through time. For example, we saw in Chapter 15 that Jules Regnault proposed and tested a stochastic process that varied randomly, which resulted in a rule about risk increasing with the square root of time.

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.” He argued that unusual events—Taleb’s black swan—were in fact much more common phenomena than Brownian motion would suggest. The crash of 1987 was not a surprise to him—he took it as a vindication of his theory. One of his major contributions to the literature on finance (published in 1966) was a proof that an efficient market implies that stock prices may not follow a random walk, but that they must be unpredictable.

pages: 193 words: 19,478

Memory Machines: The Evolution of Hypertext by Belinda Barnet


augmented reality, Benoit Mandelbrot, Bill Duvall, British Empire, Buckminster Fuller, Claude Shannon: information theory, collateralized debt obligation, computer age, conceptual framework, Douglas Engelbart, Douglas Engelbart, game design, hiring and firing, Howard Rheingold, HyperCard, hypertext link, information retrieval, Internet Archive, John Markoff, linked data, mandelbrot fractal, Marshall McLuhan, Menlo Park, nonsequential writing, Norbert Wiener, publish or perish, Robert Metcalfe, semantic web, Steve Jobs, Stewart Brand, technoutopianism, Ted Nelson, the scientific method, Vannevar Bush, wikimedia commons

—Ted Nelson, Literary Machines CONTENTS Foreword To Mandelbrot in Heaven Stuart Moulthrop Preface Chapter 1. Technical Evolution ix xix 1 Chapter 2. Memex as an Image of Potentiality 11 Chapter 3. Augmenting the Intellect: NLS 37 Chapter 4. The Magical Place of Literary Memory: Xanadu 65 Chapter 5. Seeing and Making Connections: HES and FRESS 91 Chapter 6. Machine-Enhanced (Re)minding: The Development of Storyspace 115 Conclusion 137 Notes 143 Bibliography 149 Index 157 Foreword TO MANDELBROT IN HEAVEN Stuart Moulthrop A certain confusion may befall us when we praise pioneers, especially while they are still with us. 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.

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. Understanding is catastrophic. Genius finds ‘infinite complexity […] defined by simple rules’, as Coulton also sings, though any such simplicity depends crucially on the beholder. Cosmic rules may have gorgeous clarity to a mind like Mandelbrot’s. For the rest of us, the complexities of the universe are more often bewildering.

As simple minds see it, those who light the world go to heaven before their time, and the pathos of this fate stamps the work of any chronicler with embarrassment. The singer, enraptured, invents a rapture: Mandelbrot’s in heaven – well, actually not (originally not, though he is now) – you get the idea. Time is not on our side when we try to give genius its due; we get no help, likewise, from metaphysics. We say the wrong thing, then catch ourselves in nets of qualification, tangled in the paradox of transcendence, of lives that x Memory Machines outwardly seem like ours but are actually lived on another scale, perhaps a different plane of being. Even while they breathe our ordinary air, true leaders of thought are always someplace else. We may call it heaven, and in relatively happy cases like Mandelbrot’s, maybe that’s the place. But genius can arise in any field, and local conditions differ. It is one thing to revolutionize mathematics, perhaps something else to uproot the foundations of global media, or literacy itself.

pages: 313 words: 92,053

Places of the Heart: The Psychogeography of Everyday Life by Colin Ellard

augmented reality, Benoit Mandelbrot, Berlin Wall, Broken windows theory, Buckminster Fuller, carbon footprint, commoditize, crowdsourcing, Frank Gehry, Google Glasses, Guggenheim Bilbao, haute couture, Howard Rheingold, Internet of things, Jaron Lanier, mandelbrot fractal, Marshall McLuhan, Masdar, mass immigration, megastructure, more computing power than Apollo, Oculus Rift, Peter Eisenman, RFID, Richard Florida, risk tolerance, sentiment analysis, smart cities, the built environment, theory of mind, urban decay, urban planning, urban sprawl, Victor Gruen

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. Fractal dimension describes the relationship between the length of the measuring stick and the measured length of the coastline. If the coastline happened to be a perfect straight line, its fractal dimension would be 1, so not really a fractal at all. Using mathematical tools that aren’t much different from unleashing a range of measuring sticks of different sizes on an image, it’s possible to arrive at a number that characterizes the fractal dimension of the image.

Indeed, Jackson Pollock’s paintings, though they may seem to be nothing more than a random collection of lines and splashes of color, when subjected to mathematical routines, reveal strong underlying fractal properties.14 The degree to which a visual scene is fractal in nature can be measured with any one of a number of mathematical routines that yield a number known as the scene’s fractal dimension. Understanding exactly how to interpret a given fractal dimension would take us too far into some complicated mathematics, but one way to think about this number is to think first of the dimensionality of simple geometric objects. A line has one dimension. A plane has two dimensions. A sphere has three dimensions. Fractal dimensions for scenes lie between the numbers one and two, suggesting that they are neither quite one- nor two-dimensional geometric objects. In fact, the very name “fractal” is meant to convey this property of having a fractional dimensionality lying somewhere between whole numbers.

Using mathematical tools that aren’t much different from unleashing a range of measuring sticks of different sizes on an image, it’s possible to arrive at a number that characterizes the fractal dimension of the image. When these tools are applied to scenes of nature, the measured fractal dimension often lands at a value somewhere between 1.3 and 1.5. What’s interesting about this is that psychological studies, some using a variety of scenes of nature and others using more artificial images (fractal art, abstract patterns, and even Jackson Pollock paintings), have shown that people prefer to look at images that have approximately the same range of fractal dimension as that found in nature. This correspondence between the fractal properties of images and our preference for them, and even in some cases our physiological responses to such images, which can resemble the restorative response to natural scenes, has given rise to the idea that the way that the brain actually recognizes nature is by means of this mathematical property.15 The idea of an account of our attraction to natural scenes based on the mathematics of fractals has a certain appeal.

Wireless by Stross, Charles


anthropic principle, back-to-the-land, Benoit Mandelbrot, Buckminster Fuller, Cepheid variable, cognitive dissonance, colonial exploitation, cosmic microwave background, epigenetics, finite state, Georg Cantor, gravity well, hive mind, jitney, Khyber Pass, lifelogging, Magellanic Cloud, mandelbrot fractal, peak oil, phenotype, Pluto: dwarf planet, security theater, sensible shoes, Turing machine

“Then I’m not Napoleon Bonaparte!” Oh, very droll. The terror is fading, replaced by a sense of disappointment. I trail after him. “The staff have names for you all. Turing, Cantor, Mandelbrot, and Godel. You’re not Cantor or Turing. That makes you one of Mandelbrot or Godel.” “So you’re undecided?” There’s a coffee table with a pile of newspapers on it in the middle of the dayroom, a couple of elderly chesterfields, and three armchairs that could have been looted from an old-age home sometime before the First World War. “And in any case, we haven’t been formerly introduced. So you might as well call me Alice.” Alice—or Mandelbrot or Godel or whoever he is—sits down. The armchair nearly swallows him. He beams at my bafflement, delighted to have found a new victim for his doubtless-ancient puns. “Well, Alice.

“Kurt, they sent us a tabula rasa again!” More shuffling. A stooped figure, shock-headed with white hair, appears in the doorway. He’s wearing tinted round spectacles that look like they fell off the back of a used century. “What? What?” he demands querulously. “He doesn’t know anything,” Alice confides in—this must be Godel, I realize, which means Alice is Mandelbrot—Godel, then with a wink at me, “He doesn’t know anything, either.” Godel shuffles into the restroom. “Is it teatime already?” “No!” Mandelbrot puts his mug down. “Get a watch!” “I was only asking because Alan and Georg are still playing—” This has gone far enough. Apprehension dissolves into indignation. “It’s not chess!” I point out. “And none of you are insane.” “Sssh!” Godel looks alarmed. “The Sisters might overhear!” “We’re alone, except for Dr. Renfield upstairs, and I don’t think she’s paying as much attention to what’s going on down here as she ought to.”

Obviously he’s never met a real schizophrenic. “One of you wrote a letter, alleging mistreatment by the staff. It landed on my boss’s desk, and he sent me to find out why.” THUD. Godel bounces off the wall again, showing remarkable resilience for such old bones. “Do shut up, old fellow,” chides Mandelbrot. “You’ll attract Her attention.” “I’ve met someone with K. Syndrome, and I shared a house with some real lunatics once,” I hint. “Save it for someone who cares.” “Oh bother,” says Godel, and falls silent. “We’re not mad,” Mandelbrot admits. “We’re just differently sane.” “Then why are you here?” “Public health.” He takes a sip of tea and pulls a face. “Everyone else’s health. Tell me, do they still keep an IBM 1602 in the back of the steam-ironing room?” I must look blank because he sighs deeply and subsides into his chair.

pages: 404 words: 113,514

Atrocity Archives by Stross, Charles


airport security, anthropic principle, Berlin Wall, brain emulation, British Empire, Buckminster Fuller, defense in depth, disintermediation, experimental subject, glass ceiling, haute cuisine, hypertext link, Khyber Pass, mandelbrot fractal, Menlo Park, NP-complete, the medium is the message, Y2K, yield curve

Because, you see, everything you know about the way this universe works is correct--except for the little problem that this isn't the only universe we have to worry about. Information can leak between one universe and another. And in a vanishingly small number of the other universes there are things that listen, and talk back--see Al-Hazred, Nietzsche, Lovecraft, Poe, et cetera. The many-angled ones, as they say, live at the bottom of the Mandelbrot set, except when a suitable incantation in the platonic realm of mathematics--computerised or otherwise--draws them forth. (And you thought running that fractal screen-saver was good for your computer?) Oh, and did I mention that the inhabitants of those other universes don't play by our rule book? Just solving certain theorems makes waves in the Platonic over-space. Pump lots of power through a grid tuned carefully in accordance with the right parameters--which fall naturally out of the geometry curve I mentioned, which in turn falls easily out of the Turing theorem--and you can actually amplify these waves, until they rip honking great holes in spacetime and let congruent segments of otherwiseseparate universes merge.

TWO DAYS LATER, I AM BOOKED INTO AN ORIENTATION and Objectivity seminar at the Dustbin. Only God and Bridget--and possibly Boris, though he won't say anything--know why I'm booked into an O&O course three days after getting off the plane, but something dire will probably happen if I don't turn up. The Dustbin isn't part of the Laundry, it's regular civil service, so I try to dig up a shirt that isn't too crumpled, and a tie. I own two ties--a Wile E. Coyote tie, and a Mandelbrot set tie that's particularly effective at inducing migraines--and a sports jacket that's going a bit threadbare at the cuffs. Don't want to look too out of place, do I? Someone might ask questions, and after the auto-da-fé I've just been through I do not want anyone mentioning my name in Bridget's vicinity for the next year. I'm halfway to the tube station before I remember that I forgot to shave, and I'm on the train before I notice that I'm wearing odd socks, one brown and one black.

"There's so much stuff buried in the stacks, it's unbelievable. I've been spending all my time reading, getting indigestion along the way. It's just such a waste--all that stuff, locked up behind the Official Secrets Act!" "Yeah, well." It's my turn to pull a face now. "In principle, I kind of agree with you. In practice . . . how to put it? This stuff has repercussions. The many-angled ones live at the bottom of the Mandelbrot set; play around with it for too long and horrible things can happen to you." I shrug. "And you know what students are like." "Yes, well." She stands up, straightening her skirt with one hand and holding the book with the other. "I suppose you've got more experience of that than I have. But, well." She pauses, and gives a little halfsmile: "I was wondering if, if you'd eaten yet?" Ah.

pages: 543 words: 147,357

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


Andrei Shleifer, asset-backed security, bank run, banking crisis, Benoit Mandelbrot, Berlin Wall, Bernie Madoff, Big bang: deregulation of the City of London, Bretton Woods, capital controls, carbon footprint, Carmen Reinhart, Cass Sunstein, centre right, choice architecture, cloud computing, collective bargaining, conceptual framework, Corn Laws, corporate governance, creative destruction, credit crunch, Credit Default Swap, debt deflation, decarbonisation, Deng Xiaoping, discovery of DNA, discovery of the americas, discrete time, diversification, double helix, Edward Glaeser, financial deregulation, financial innovation, financial intermediation, first-past-the-post, floating exchange rates, Francis Fukuyama: the end of history, Frank Levy and Richard Murnane: The New Division of Labor, full employment, George Akerlof, Gini coefficient, global supply chain, Growth in a Time of Debt, Hyman Minsky, I think there is a world market for maybe five computers, income inequality, inflation targeting, interest rate swap, invisible hand, Isaac Newton, James Dyson, James Watt: steam engine, joint-stock company, Joseph Schumpeter, Kenneth Rogoff, knowledge economy, knowledge worker, labour market flexibility, liberal capitalism, light touch regulation, Long Term Capital Management, Louis Pasteur, low-wage service sector, mandelbrot fractal, margin call, market fundamentalism, Martin Wolf, mass immigration, means of production, Mikhail Gorbachev, millennium bug, money market fund, moral hazard, moral panic, mortgage debt, Myron Scholes, Neil Kinnock, new economy, Northern Rock, offshore financial centre, open economy, Plutocrats, plutocrats, price discrimination, private sector deleveraging, purchasing power parity, quantitative easing, race to the bottom, railway mania, random walk, rent-seeking, reserve currency, Richard Thaler, Right to Buy, rising living standards, Robert Shiller, Robert Shiller, Ronald Reagan, Rory Sutherland, Satyajit Das, shareholder value, short selling, Silicon Valley, Skype, South Sea Bubble, Steve Jobs, The Market for Lemons, the market place, The Myth of the Rational Market, the payments system, the scientific method, The Wealth of Nations by Adam Smith, too big to fail, unpaid internship, value at risk, Vilfredo Pareto, Washington Consensus, wealth creators, working poor, zero-sum game, éminence grise

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’. Mandelbrot withdrew from economics to ask the same questions in the natural sciences.38 Forty-five years later, we have the assurance that Cootner demanded. But even after the earthquake too few are fessing up to the awesomeness of their mistake.

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: 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 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.

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.

pages: 545 words: 137,789

How Markets Fail: The Logic of Economic Calamities by John Cassidy


Albert Einstein, Andrei Shleifer, anti-communist, asset allocation, asset-backed security, availability heuristic, bank run, banking crisis, Benoit Mandelbrot, Berlin Wall, Bernie Madoff, Black-Scholes formula, Bretton Woods, British Empire, capital asset pricing model, centralized clearinghouse, collateralized debt obligation, Columbine, conceptual framework, Corn Laws, corporate raider, correlation coefficient, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, crony capitalism, Daniel Kahneman / Amos Tversky, debt deflation, diversification, Elliott wave, Eugene Fama: efficient market hypothesis, financial deregulation, financial innovation, Financial Instability Hypothesis, financial intermediation, full employment, George Akerlof, global supply chain, Gunnar Myrdal, Haight Ashbury, hiring and firing, Hyman Minsky, income per capita, incomplete markets, index fund, information asymmetry, Intergovernmental Panel on Climate Change (IPCC), invisible hand, John Nash: game theory, John von Neumann, Joseph Schumpeter, Kenneth Arrow, laissez-faire capitalism, Landlord’s Game, liquidity trap, London Interbank Offered Rate, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, market bubble, market clearing, mental accounting, Mikhail Gorbachev, money market fund, Mont Pelerin Society, moral hazard, mortgage debt, Myron Scholes, Naomi Klein, negative equity, Network effects, Nick Leeson, Northern Rock, paradox of thrift, Pareto efficiency, Paul Samuelson, Ponzi scheme, price discrimination, price stability, principal–agent problem, profit maximization, quantitative trading / quantitative finance, race to the bottom, Ralph Nader, RAND corporation, random walk, Renaissance Technologies, rent control, Richard Thaler, risk tolerance, risk-adjusted returns, road to serfdom, Robert Shiller, Robert Shiller, Ronald Coase, Ronald Reagan, shareholder value, short selling, Silicon Valley, South Sea Bubble, sovereign wealth fund, statistical model, technology bubble, The Chicago School, The Great Moderation, The Market for Lemons, The Wealth of Nations by Adam Smith, too big to fail, transaction costs, unorthodox policies, value at risk, Vanguard fund, Vilfredo Pareto, wealth creators, zero-sum game

Bernstein, Capital Ideas: The Improbable Origins of Modern Wall Street (New York: Free Press, 1993), 17–18. 88 “[t]he mathematical expectation . . .”: Quoted in ibid., 21. 88 “Suppose you see . . .”: Benoit Mandelbrot and Richard Hudson, The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward (New York: Basic Books, 2006), 52. 89 “even if his powers . . .”: Quoted in Bernstein, Capital Ideas, 134. 90 Fama’s follow-up paper: Eugene Fama, “Efficient Capital Markets: A Review of Theory and Empirical Work,” Journal of Finance 25, no. 2 (1970): 383–417; summarized in Bernstein, Capital Ideas, 137–38. 90 “The past history of stock prices . . .”: Burton Gordon Malkiel, A Random Walk Down Wall Street: The Best and Latest Investment Advice Money Can Buy, 6th ed. (New York: W. W. Norton, 1996), 161. 94 “If the cotton-price changes fit . . .”: Mandelbrot and Hudson, (Mis)behavior of Markets, 168. 94 “In fact, the bell curve . . .”: Ibid., 13. 95 “[L]arge changes tend to . . .”: Ibid., 248. 96 “Modern finance was the . . .”: Ibid., 167. 8.

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. The Harvard economist Hendrik Houthakker, whom he met while giving a talk in Cambridge, gave him the records of daily movements in the prices of cotton and cotton futures going back more than a century, which he had obtained from the New York Cotton Exchange. Mandelbrot plotted the changes in prices on a computer and quickly saw that they didn’t look anything like a smooth bell curve.

“If the cotton-price changes fit the standard theory, they would be like sand grains in a heap: somewhat different sizes, but all sand grains, nonetheless,” Mandelbrot and a coauthor, Richard Hudson, recalled in their 2004 book, The (Mis)Behavior of Markets. “But the cotton research showed something different . . . Some days, cotton prices hardly budged from the previous close; those are the small sand grains . . . Other days—perhaps word of a drought in Missouri finally reached New York—the news was big: wild price moves, statistical boulders.” Mandelbrot’s evidence was confined to an obscure corner of the financial market, but other researchers subsequently found similar discrepancies in the behavior of many other speculative assets, including stocks, bonds, and currencies. “In fact, the bell curve fits reality very poorly,” Mandelbrot and Hudson note. “From 1916 to 2003, the daily index movements of the Dow Jones Industrial Average do not spread out on graph paper like a simple bell curve.

pages: 467 words: 154,960

Trend Following: How Great Traders Make Millions in Up or Down Markets by Michael W. Covel


Albert Einstein, asset allocation, Atul Gawande, backtesting, beat the dealer, Bernie Madoff, Black Swan, buy low sell high, capital asset pricing model, Clayton Christensen, commodity trading advisor, computerized trading, correlation coefficient, Daniel Kahneman / Amos Tversky, delayed gratification, deliberate practice, diversification, diversified portfolio, Edward Thorp, Elliott wave, Emanuel Derman, Eugene Fama: efficient market hypothesis, Everything should be made as simple as possible, fiat currency, fixed income, game design, hindsight bias, housing crisis, index fund, Isaac Newton, John Meriwether, John Nash: game theory, linear programming, Long Term Capital Management, mandelbrot fractal, margin call, market bubble, market fundamentalism, market microstructure, mental accounting, money market fund, Myron Scholes, Nash equilibrium, new economy, Nick Leeson, Ponzi scheme, prediction markets, random walk, Renaissance Technologies, Richard Feynman, Richard Feynman, risk tolerance, risk-adjusted returns, risk/return, Robert Shiller, Robert Shiller, shareholder value, Sharpe ratio, short selling, South Sea Bubble, Stephen Hawking, survivorship bias, systematic trading, the scientific method, Thomas L Friedman, too big to fail, transaction costs, upwardly mobile, value at risk, Vanguard fund, volatility arbitrage, William of Occam, zero-sum game

Roger Lowenstein, Wall Street Journal, June 13, 2003. 20. Benoit B. Mandelbrot, A Multifractal Walk Down Wall Street. Scientific American, Vol. 280, No. 2 (February 1999), 70–73. 21. Larry Swedroe, Buckingham Asset Management. See http://www. 22. Larry Swedroe, Buckingham Asset Management. See http://www. 23. Mark Rzepczynski, Ph.D., Return Distribution Properties of JWH Investment Programs, Stock and Bond Indices, and Hedge Funds. John W. Henry and Co., No. V, June 2000. 24. National Institute of Standards and Technology. See www.itl.nist. gov. 25. National Institute of Standards and Technology. See www.itl.nist. gov. 26. Jim Rogers, Investment Biker. New York: Random House, 1994. 27. Larry S. Liebovitch, Two Lessons from Fractals and Chaos. Complexity, Vol. 5, No. 4 (2000), 34–43. 28.

The risk-reducing formulas behind portfolio theory rely on a number of demanding and ultimately unfounded premises. First, they suggest that price changes are statistically independent from one another…The second assumption is that price changes are distributed in a pattern that conforms to a standard bell curve. Do financial data neatly conform to such assumptions? Of course, they never do. Benoit B. Mandelbrot20 Trend Following (Updated Edition): Learn to Make Millions in Up or Down Markets higher or lower than that commonly associated with a normal distribution. For example, a return series of –30 percent, 5 percent, 10 percent, and 15 percent has a mean of 0 percent. Only one return is less than 0 percent, whereas three are higher but the one that is negative is much farther from the mean (0 percent) than the positive ones.

., Jr, 62 Kerkorian, Kirk, 111 Killian, Mike, 125 Kingman, Dave, 143, 183-184 Klingler, James, 107 Klopenstein, Ralph, 39 Knapp, Volker, 393 Knoepffler, Alejandro, 273 Koppel, Ted, 117 Kovner, Bruce, 62, 282, 285, 289 Kozloff, Burt, 16 Kroc, Ray, 377 Kurczek, Dion, 393 kurtosis (statistics), 228 Lange, Harry, 111 Lao Tsu, 195 “law of small numbers,” 195 Le Bon, Gustave, 201 leadership traits, 201 “Learning to Love Non-Correlation” (research paper), 112 Lector, Hannibal, 221 Lee Kuan Yew, 205 Lee, Sang, 125 Leeson, Nick, 124-125, 168-172 Lefevre, Edwin, 91 Legg Mason, 285-286 Leggett, Robert, 241 Lehman Brothers, 153 Leonardo da Vinci, 242 leverage, decreasing returns and, 281-282 Levine, Karen, 203 Lewis, Michael, 184, 188 Liechtenstein Global Trust, 156 limitations of day trading, 272 linear versus nonlinear world, 224-229 Litner, John, 86 Little, Grady, 188-189 Little, Jim, 69, 71, 151, 253, 261 Litvinenko, Alexander, 199 Livermore, Jesse, 22, 90-93, 131, 236 Lo, Andrew, 271 Lombardi, Vince, 65, 176 Long Island Business News, 375 Long Term Capital Management (LTCM), xix, 118, 151-164, 272, 280, 293 long volatility, defined, 422 losers averaging, 235, 237-238 winners versus, 123-125 losing investment philosophies, 4-6 losing positions, when to exit, 262-263 losses. See also drawdowns handling, 22-23, 195-196 Long-Term Capital Management (LTCM) collapse, 156 zero-sum trading, 114-120 lottery example (risk and reward), 250-251 Lowenstein, Roger, 259 Lueck, Martin, 29 lumber trading, 134 Lynch, Peter, 110 Madoff, Bernard, 22, 223 The Man Group, 15, 29, 148, 157-158, 287 Managed Account Reports, 376 Mandelbrot, Benoit B., 228 manias, prospect theory, 194-199 Marcus, Michael, 19, 60, 62, 285 Marino, Dan, 261 market defined, 3-4 inefficiency of, 288-290 role of speculation in, 6 market price. See price market theories fundamental analysis, 7-9 technical analysis, 9-11 Market Wizards (Schwager), xvi, 58 Markowitz, Harry, xx, 86 Martin, Michael, 64 435 Index Martinez, Pedro, 186, 188 Mauboussin, Michael, 124, 181, 218, 226, 286 McCann, Timothy, 285 McCarver, Tim, 214 Meaden, Nicola, 102 Mechanica software, 385-393 mechanical trading systems, 11-12 Melamed, Leo, 123, 273-274 Memos from the Chairman (Greenberg), 205 Mencken, H.

pages: 425 words: 122,223

Capital Ideas: The Improbable Origins of Modern Wall Street by Peter L. Bernstein


Albert Einstein, asset allocation, backtesting, Benoit Mandelbrot, Black-Scholes formula, Bonfire of the Vanities, Brownian motion, buy low sell high, capital asset pricing model, corporate raider, debt deflation, diversified portfolio, Eugene Fama: efficient market hypothesis, financial innovation, financial intermediation, fixed income, full employment, implied volatility, index arbitrage, index fund, interest rate swap, invisible hand, John von Neumann, Joseph Schumpeter, Kenneth Arrow, law of one price, linear programming, Louis Bachelier, mandelbrot fractal, martingale, means of production, money market fund, Myron Scholes, new economy, New Journalism, Paul Samuelson, profit maximization, Ralph Nader, RAND corporation, random walk, Richard Thaler, risk/return, Robert Shiller, Robert Shiller, Ronald Reagan, stochastic process, the market place, The Predators' Ball, the scientific method, The Wealth of Nations by Adam Smith, Thorstein Veblen, transaction costs, transfer pricing, zero-coupon bond, zero-sum game

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.

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. Mandelbrot’s research implied that stocks are riskier than had been assumed, that diversification might not work as well as Markowitz had indicated, that measures like variance could be highly unstable, and that major price movements would cluster more closely than anticipated. Mandelbrot’s view of the stock market was the genesis of what is known today as Chaos Theory, of which Mandelbrot himself is an articulate proponent. The events of October 1987 and less dramatic but qualitatively similar episodes lend some credence to Mandelbrot’s warnings.

The events of October 1987 and less dramatic but qualitatively similar episodes lend some credence to Mandelbrot’s warnings. Despite those events, however, Mandelbrot remains on the periphery of financial theory, both because of the inconvenience to analysts of accepting his arguments and because of the natural human desire to hope that fluctuations will remain within familiar bounds. ••• Soon after his foray into the feverish world described by Mandelbrot, Fama turned to full analysis of the random behavior of stock prices. In January 1965, the Journal of Business published his article “The Behavior of Stock Market Prices.” Seventy pages long, it comprised his entire Ph.D. thesis—an extraordinary compliment from the editors of a leading journal to a young man who was just beginning his career.

pages: 396 words: 117,149

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


3D printing, Albert Einstein, Amazon Mechanical Turk, Arthur Eddington, basic income, Bayesian statistics, Benoit Mandelbrot, bioinformatics, Black Swan, Brownian motion, cellular automata, Claude Shannon: information theory, combinatorial explosion, computer vision, constrained optimization, correlation does not imply causation, creative destruction, crowdsourcing, Danny Hillis, data is the new oil, double helix, Douglas Hofstadter, Erik Brynjolfsson, experimental subject, Filter Bubble, future of work, global village, Google Glasses, Gödel, Escher, Bach, information retrieval, job automation, John Markoff, John Snow's cholera map, John von Neumann, Joseph Schumpeter, Kevin Kelly, lone genius, mandelbrot fractal, Mark Zuckerberg, Moneyball by Michael Lewis explains big data, Narrative Science, Nate Silver, natural language processing, Netflix Prize, Network effects, NP-complete, off grid, P = NP, PageRank, pattern recognition, phenotype, planetary scale, pre–internet, random walk, Ray Kurzweil, recommendation engine, Richard Feynman, Richard Feynman, Second Machine Age, self-driving car, Silicon Valley, speech recognition, statistical model, Stephen Hawking, Steven Levy, Steven Pinker, superintelligent machines, the scientific method, The Signal and the Noise by Nate Silver, theory of mind, Thomas Bayes, transaction costs, Turing machine, Turing test, Vernor Vinge, Watson beat the top human players on Jeopardy!, white flight, zero-sum game

Otto Creutzfeldt makes the case that the cortex is one algorithm in “Generality of the functional structure of the neocortex” (Naturwissenschaften, 1977), as does Vernon Mountcastle in “An organizing principle for cerebral function: The unit model and the distributed system,” in The Mindful Brain, edited by Gerald Edelman and Vernon Mountcastle (MIT Press, 1978). 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). The Golden Ticket, by Lance Fortnow (Princeton University Press, 2013), is an introduction to NP-completeness and the P = NP problem. The Annotated Turing,* by Charles Petzold (Wiley, 2008), explains Turing machines by revisiting Turing’s original paper on them.

Countless wrong inferences could be drawn from those observations, but most of them never occur to us, because our inferences are influenced by our broad knowledge of the world, and that knowledge is consistent with the laws of nature. How much of the character of physical law percolates up to higher domains like biology and sociology remains to be seen, but the study of chaos provides many tantalizing examples of very different systems with similar behavior, and the theory of universality explains them. The Mandelbrot set is a beautiful example of how a very simple iterative procedure can give rise to an inexhaustible variety of forms. If the mountains, rivers, clouds, and trees of the world are all the result of such procedures—and fractal geometry shows they are—perhaps those procedures are just different parametrizations of a single one that we can induce from them. In physics, the same equations applied to different quantities often describe phenomena in completely different fields, like quantum mechanics, electromagnetism, and fluid dynamics.

., 34–38 Machine learning, 6–10 analogy and, 178–179 bias and variance and, 78–79 big data and, 15–16 business and, 10–13 chunking, 223–227 clustering, 205–210 dimensionality reduction, 211–217 effect on employment, 276–279 exponential function and, 73–74 fitness function and, 123 further readings, 297–298 future of, 21–22 impact on daily life, 298 effect on employment, 276–279 meta-learning, 237–239 nature vs. nurture debate and, 29, 137–139 Newton’s principle and, 65–66 planetary-scale, 256–259 politics and, 16–19 principal-component analysis, 211–217 problem of unpredictability and, 38–40 reinforcement learning, 218–223, 226–227 relational learning, 227–233 relationship to artificial intelligence, 8 science and, 13–16, 235–236 significance tests and, 76–77 as technology, 236–237 Turing point and, 286, 288 war and, 19–21, 279–282 See also Algorithms Machine-learning problem, 61–62, 109–110 Machine-translation systems, 154 MacKay, David, 170 Madrigal, Alexis, 273–274 Malthus, Thomas, 178, 235 Manchester Institute of Biotechnology, 16 Mandelbrot set, 30, 300 Margins, 192–194, 196, 241, 242, 243, 307 Markov, Andrei, 153 Markov chain Monte Carlo (MCMC), 164–165, 167, 170, 231, 241, 242, 253, 256 Markov chains, 153–155, 159, 304–305 Markov logic. See Markov logic networks (MLNs) Markov logic networks (MLNs), 246–259, 309–310 classes and, 257 complexity and, 258–259 parts and, 256–257 with hierarchical structure, 256–257 See also Alchemy Markov networks, 171–172, 229, 240, 245, 253, 306 Marr, David, 89 Marr’s three levels, 89 Master Algorithm, 239–246 Alchemy and, 250–259 Bayes’ theorem and, 148 brain as, 26–28 CanceRx, 259–261 candidates that fail as, 48–50 chunking and, 226 complexity of, 40–41 as composite picture of current and future learners, 263–264 computer science and, 32–34 equation, 50 evolution and, 28–29 five tribes and, 51–55 future and, 292 goal of, 39 Google and, 282 invention of, 25–26 Markov logic networks and, 236–250 meta-learning and, 237–239 physics and, 29–31 practical applications of, 41–45 statistics and, 31–32 symbolism and, 90–91 theory of everything and, 46–48 Turing point and, 286, 288 as unifier of machine learning, 237 unity of knowledge and, 31, 12, 265 Matrix factorization for recommendation systems, 215 Maximum likelihood principle, 166–167, 168 Maxwell, James Clerk, 235 McCulloch, Warren, 96 McKinsey Global Institute, 9 MCMC.

pages: 999 words: 194,942

Clojure Programming by Chas Emerick, Brian Carper, Christophe Grand


Amazon Web Services, Benoit Mandelbrot, cloud computing, continuous integration, database schema, domain-specific language, don't repeat yourself,, failed state, finite state, Firefox, game design, general-purpose programming language, Guido van Rossum, Larry Wall, mandelbrot fractal, Paul Graham, platform as a service, premature optimization, random walk, Ruby on Rails, Schrödinger's Cat, semantic web, software as a service, sorting algorithm, Turing complete, type inference, web application

[348] This approach was originally described in Visualizing the Mandelbrot Set in Clojure Let’s take a look at a somewhat more interesting example than the overused Fibonacci and prime number generators that are often used for microbenchmarking numeric performance. Visualizing the Mandelbrot Set[349] (or really, any fractal shape visualization) has long been a common practicum, and it will serve well here as a demonstration of how to optimize numeric algorithms in Clojure. The Mandelbrot Set is defined by a complex polynomial that is applied iteratively: zk+1 = zk2 + c where c (a complex number) is a member of the Mandelbrot Set if zk+1 is bounded as k increases when z0 is initialized to 0. c’s that produce unbounded results from this calculation are said to escape to infinity.

y) (partition width escapes) (recur 0 (dec y) escapes)) (recur (inc x) y (conj escapes (escape (+ rmin (* x stride-w)) (+ imin (* y stride-h)) depth))))))) (defn render-text "Prints a basic textual rendering of mandelbrot set membership, as returned by a call to `mandelbrot`." [mandelbrot-grid] (doseq [row mandelbrot-grid] (doseq [escape-iter row] (print (if (neg? escape-iter) \* \space))) (println))) (defn render-image "Given a mandelbrot set membership grid as returned by a call to `mandelbrot`, returns a BufferedImage with the same resolution as the grid that uses a discrete grayscale color palette." [mandelbrot-grid] (let [palette (vec (for [c (range 500)] (Color/getHSBColor 0.0 0.0 (/ (Math/log c) (Math/log 500))))) height (count mandelbrot-grid) width (count (first mandelbrot-grid)) img (BufferedImage. width height BufferedImage/TYPE_INT_RGB) ^java.awt.Graphics2D g (.getGraphics img)] (doseq [[y row] (map-indexed vector mandelbrot-grid) [x escape-iter] (map-indexed vector row)] (.setColor g (if (neg?

First, let’s look at a naive implementation of the Mandelbrot Set in Clojure,[350] which includes a couple utility functions for rendering the results of that implementation: Example 11-4. Mandelbrot Set in Clojure (ns clojureprogramming.mandelbrot (:import java.awt.image.BufferedImage (java.awt Color RenderingHints))) (defn- escape "Returns an integer indicating how many iterations were required before the value of z (using the components `a` and `b`) could be determined to have escaped the Mandelbrot set. If z will not escape, -1 is returned." [a0 b0 depth] (loop [a a0 b b0 iteration 0] (cond (< 4 (+ (* a a) (* b b))) iteration (>= iteration depth) -1 :else (recur (+ a0 (- (* a a) (* b b))) (+ b0 (* 2 (* a b))) (inc iteration))))) (defn mandelbrot "Calculates membership within and number of iterations to escape from the Mandelbrot set for the region defined by `rmin`, `rmax` `imin` and `imax` (real and imaginary components of z, respectively).

pages: 323 words: 95,939

Present Shock: When Everything Happens Now by Douglas Rushkoff


algorithmic trading, Andrew Keen, bank run, Benoit Mandelbrot, big-box store, Black Swan, British Empire, Buckminster Fuller, cashless society, citizen journalism, clockwork universe, cognitive dissonance, Credit Default Swap, crowdsourcing, Danny Hillis, disintermediation, Donald Trump, double helix, East Village, Elliott wave, European colonialism, Extropian, facts on the ground, Flash crash, game design, global supply chain, global village, Howard Rheingold, hypertext link, Inbox Zero, invention of agriculture, invention of hypertext, invisible hand, iterative process, John Nash: game theory, Kevin Kelly, laissez-faire capitalism, Law of Accelerating Returns, loss aversion, mandelbrot fractal, Marshall McLuhan, Merlin Mann, Milgram experiment, mutually assured destruction, negative equity, Network effects, New Urbanism, Nicholas Carr, Norbert Wiener, Occupy movement, passive investing, pattern recognition, peak oil, price mechanism, prisoner's dilemma, Ralph Nelson Elliott, RAND corporation, Ray Kurzweil, recommendation engine, selective serotonin reuptake inhibitor (SSRI), Silicon Valley, Skype, social graph, South Sea Bubble, Steve Jobs, Steve Wozniak, Steven Pinker, Stewart Brand, supply-chain management, the medium is the message, The Wisdom of Crowds, theory of mind, Turing test, upwardly mobile, Whole Earth Catalog, WikiLeaks, Y2K, zero-sum game

Many of the “quant” teams at hedge funds and the risk-management groups within brokerage houses use fractals to find technical patterns in stock market movements. 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.

It’s a sensibility we find reinforced by systems theory and chaos math. Fractals (those computer-rendered topologies that were to early cyberculture what paisley was to the 1960s) help us make sense of rough, natural phenomena, everything from clouds and waves to rocks and forests. Unlike traditional, Euclidean mathematics, which has tended to smooth out complexity, reducing it down to oversimplified lines and curves, fractal geometry celebrates the way real objects aren’t really one, two, or three dimensions, but ambiguously in between. Fractals are really just recursive equations—iterations upon iterations of numbers. But when they are rendered by computers, they churn out beautiful, complex patterns. They can look like a coral reef or a fern or a weather system. What makes fractals so interesting is that they are self-similar.

On the one hand, this makes fractals terrifically orienting: as above, so below. Nature is patterned, which is part of what makes a walk in the woods feel reassuring. The shapes of the branches are reflected in the veins of the leaves and the patterns of the paths between the trunks. The repeating patterns in fractals also seem to convey a logic or at least a pattern underlying the chaos. On the other hand, once you zoom in to a fractal, you have no way of knowing which level you are on. The details at one level of magnification may be the same as on any other. Once you dive in a few levels, you are forever lost. Like a dream within a dream within a dream (as in the movie Inception), figuring out which level you are on can be a challenge, or even futile. Meanwhile, people are busy using fractals to explain any system that has defied other, more reductionist approaches.

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


Antoine Gombaud: Chevalier de Méré, availability heuristic, backtesting, Benoit Mandelbrot, Black Swan, commoditize, complexity theory, corporate governance, corporate raider, currency peg, Daniel Kahneman / Amos Tversky, discounted cash flows, diversified portfolio, endowment effect, equity premium, fixed income, global village, hindsight bias, Kenneth Arrow, Long Term Capital Management, loss aversion, mandelbrot fractal, mental accounting, meta analysis, meta-analysis, Myron Scholes, Paul Samuelson, quantitative trading / quantitative finance, QWERTY keyboard, random walk, Richard Feynman, Richard Feynman, road to serfdom, Robert Shiller, Robert Shiller, selection bias, shareholder value, Sharpe ratio, Steven Pinker, stochastic process, survivorship bias, too big to fail, Turing test, Yogi Berra

Raiffa, 1957, Games and Decisions: Introduction and Critical Survey. New York: Dover. Machina, M. J., and M. Rothschild, 1987, “Risk.” In Eatwell, J., Milgate, M., and Newman P., eds., 1987, The New Palgrave: A Dictionary of Economics. London: Macmillan. MacKay, Charles, 2002, Extraordinary Popular Delusions and the Madness of Crowds. New York: Metro Books. Magee, Bryan, 1997, Confessions of a Philosopher. London: Weidenfeld & Nicholson. Mandelbrot, Benoit B., 1997, Fractals and Scaling in Finance. New York: Springder-Verlag. Markowitz, Harry, 1959, Portfolio Selection: Efficient Diversification of Investments, 2nd ed. New York: Wiley. Meehl, Paul E., 1954, Clinical Versus Statistical Predictions: A Theoretical Analysis and Revision of the Literature. Minneapolis: University of Minnesota Press. Menand, Louis, 2001, The Metaphysical Club: A Story of Ideas in America.

Are these “critical points” not quite points but progressions (the so-called Pareto power laws)? 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. Our emotional apparatus is designed for linear causality.

The reasons economists never liked to use it is that it does not offer tractable properties—economists like to write papers in which they offer the illusion of solutions, particularly in the form of mathematical answers. A Pareto-Levy distribution does not provide them with such luxury. For economic discussions on the ideas of Pareto, see Zajdenweber (2000), Bouvier (1999). For a presentation of the mathematics of Pareto-Levy distributions, see Voit (2001), and Mandelbrot (1997). There is a recent rediscovery of power law dynamics. Intuitively a power law distribution has the following property: If the power exponent were 2, then there would be 4 times more people with an income higher than $1 million than people with $2 million. The effect is that there is a very small probability of having an event of an extremely large deviation. More generally given a deviation x, the incidence of a deviation of a multiple of x will be that multiple to a given power exponent.

pages: 443 words: 51,804

Handbook of Modeling High-Frequency Data in Finance by Frederi G. Viens, Maria C. Mariani, Ionut Florescu


algorithmic trading, asset allocation, automated trading system, backtesting, Black-Scholes formula, Brownian motion, business process, continuous integration, corporate governance, discrete time, distributed generation, fixed income, Flash crash, housing crisis, implied volatility, incomplete markets, linear programming, mandelbrot fractal, market friction, market microstructure, martingale, Menlo Park, p-value, pattern recognition, performance metric, principal–agent problem, random walk, risk tolerance, risk/return, short selling, statistical model, stochastic process, stochastic volatility, transaction costs, value at risk, volatility smile, Wiener process

Phys Rev E 1995;52:1197–1199. 25. Podobnik B, Ivanov PCh, Lee Y, Stanley HE. Scale-invariant truncated Levy process. Europhys Lett 2000;52:491–497. 26. Shiryaev AN. Essentials of the stochastic finance. World Scientific, Hackensack, New Jersey; 2008. 27. Hurst HE. Long term storage of reservoirs. Trans Am Soc Civ Eng 1950;116:770–808. 28. Mandelbrot BB, Van Ness JW. Fractional Brownian motions, fractional noises and applications. SIAM Rev 1968;10(4): 422–437. 29. Mandelbrot BB. The fractal geometry of nature. New York: Freeman and Co.; 1982. 30. Ivanova K, Ausloos M. Application of the Detrended Fluctuation Analysis (DFA) method for describing cloud breaking. Physica A 1999;274:349–354. 162 CHAPTER 6 Long Correlations Applied to the Study of Memory 31. Buldyrev SV, Buldyrev SV, Goldberger AL, Havlin S, Mantegna RN, Matsa ME, Peng CK, Simons M, Stanley HE.

The first model that described the evolution of option prices was the Brownian motion. This model assumes that the increment of the logarithm of prices follows a diffusive process with Gaussian distribution [12]. However, the empirical study of temporal series of some of the most important indices shows that in short time intervals, the associated pdfs have greater kurtosis than a Gaussian distribution [5]. The first step to explain this behavior was done in 1963 by Mandelbrot [13]. He developed a model for the evolution of cotton prices by a stable stochastic non-Gaussian Levy process; these types of non-Gaussian processes were first introduced and studied by Levy [14]. The other major problem encountered in the analysis of the behavior of different time-series data is the existence of long-term or short-term correlations in the behavior of financial markets (established versus emerging markets [15], developed countries’ market indices [1–5], Bombay stock exchange index [16], Latin American indices [17], and the references therein).

Furthermore, the TLF maintains statistical properties that are indistinguishable from the Levy flights [15]. 6.2.2 RESCALED RANGE ANALYSIS Hurst [27] initially developed the Rescaled range analysis (R/S analysis). He observed many natural phenomena that followed a biased random walk, that is, every phenomenon showed a pattern. He measured the trend using an exponent now called the Hurst exponent. Mandelbrot [28,29] later introduced a generalized form of the Brownian motion model, the fractional Brownian motion to model the Hurst effect. The numerical procedure to estimate the Hurst exponent H by using the R/S analysis is presented next (for more details, please see [27] and references therein). 1. Let N be the length of time series (y1 , y2 , y3 , . . . , yN ). The logarithmic ratio of the time series is obtained.

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


additive manufacturing, Atul Gawande, backtesting, Benoit Mandelbrot, buy low sell high, Checklist Manifesto, computerized trading, deliberate practice, diversification, Elliott wave, endowment effect, loss aversion, mandelbrot fractal, margin call, offshore financial centre, paper trading, Ponzi scheme, price stability, psychological pricing, quantitative easing, random walk, risk tolerance, short selling, South Sea Bubble, systematic trading, The Wisdom of Crowds, transaction costs, transfer pricing, traveling salesman, tulip mania, zero-sum game

Your system should recognize this transition, and that's when you should put on a trade! Earlier we spoke about the one great advantage of a private trader over professionals—he may wait for a good trade instead of having to be active each day. The chaos theory confirms that message. The chaos theory also teaches us that orderly structures that emerge from chaos are fractal. The sea coast appears equally jagged whether you look down on it from space or an airplane, from a standing position or on your knees through a magnifying glass. Market patterns are fractal as well. If I show you a set of charts of the same market, having removed time markings, you will not be able to tell whether it is monthly, weekly, daily, or a 5-minute chart. Later in this book (Chapter 39), we'll return to this theme, and you'll see why it is so important to analyze markets in more than one timeframe.

Resistance is where sellers sell with greater intensity than buyers buy (see Chapter 18). 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.

on CFDs cutting of former institutional traders inability to manage on options per share, limiting psychological effect of 6% Rule to limit 2% Rule to limit Loss aversion Lovvorn, Kerry Low-priced stocks, indictors based on volume of “Low” volume M MAs, see Moving averages MACD, see Moving Average Convergence-Divergence MACD-Histogram combined with channels divergences in Impulse system and market psychology peaks and valleys seasons of semiautomatic divergence scanner slope of time windows of trading rules in Triple Screen system MACD Lines crossover of Signal lines and MACD line in divergences and market psychology trading rules Mackay, Charles MacMillan, Lawrence Magic method gurus Managing trades forecasting vs. and poll-taking by reading markets and managing yourself Mandelbrot, Benoit Margins Margin calls Market(s): attempts to manipulate and automatic trading systems comparing volumes of contango as crowds. See also Mass psychology crowd mentality experts on independent thinking vs. of individuals leaders of crowds reasons for joining crowds wisdom of crowds ETFs groups vs. individuals in harshness of inability to control inside information in overbought and oversold randomness in reading seasons of size of source of money in spikes in as sport theories of timeframes of analysis using multiple timeframes conflicting trading ranges vs. trends in worldwide crowds Market cycle gurus Market data: in computerized technical analysis in moving averages Market indexes, in technical analysis Market makers Market noise: perceived cycles as and placement of stops setting stops outside zone of Market orders bid-ask spreads for slippage on Market panics Market participants, groups of Market tide screen (Triple Screen system) Market time Market Vane Market wave screen (Triple Screen trading system) MAS (Most Active Stocks) indicator Mass manias Mass psychology and emergence of gurus managing trades forecasting vs.

pages: 313 words: 34,042

Tools for Computational Finance by Rüdiger Seydel


bioinformatics, Black-Scholes formula, Brownian motion, commoditize, continuous integration, discrete time, implied volatility, incomplete markets, interest rate swap, linear programming, London Interbank Offered Rate, mandelbrot fractal, martingale, random walk, stochastic process, stochastic volatility, transaction costs, value at risk, volatility smile, Wiener process, zero-coupon bond

Lyuu: Financial Engineering and Computation. Principles, Mathematics, Algorithms. Cambridge University Press, Cambridge (2002). L.W. MacMillan: Analytic approximation for the American put option. Advances in Futures and Options Research 1 (1986) 119-139. R. Mainardi, M. Roberto, R. Gorenflo, E. Scalas: Fractional calculus and continuous-time finance II: the waiting-time distribution. Physica A 287 (2000) 468-481. B.B. Mandelbrot: A multifractal walk down Wall Street. Scientific American, Febr. 1999, 50–53. M. Marchesi, S. Cinotti, S. Focardi, M. Raberto: Development and testing of an artificial stock market. Proceedings Urbino 2000, Ed. I.-G. Bischi (2000). References [Mar78] [Ma68] [MaB64] [Mas99] [MaPS02] [MaN98] [Mayo00] [McW01] [MeVN02] [Mer73] [Mer76] [Me90] [Mik98] [Mi74] [Moe76] [Moro95] [MC94] [Mo98] [Mo96] [MR97] [Ne96] [New97] [Ni78] [Ni92] 289 W.

For illustration assume that y0 (for t = 0) and yT (for t = T ) are to be connected. Then the Brownian bridge is defined by t t t Bt = y0 1 − + yT + Wt − WT . T T T The first two terms represent a straight-line connection between y0 and yT . This straight line stands for the trend. The term Wt − Tt WT describes the stochastic fluctuation. For its realization an appropriate volatility can be prescribed (−→ Exercise 3.7). Another alternative to fill large gaps is to apply fractal interpolation [Man99]. 3.5 Monte Carlo Simulation As pointed out in Section 2.4 in the context of calculating integrals, Monte Carlo is attractive in high-dimensional spaces. The same characterization holds when Monte Carlo is applied to the valuation of options. For sake of clarity we describe the approach in the one-dimensional context. From Section 1.7.2 we take the one-factor model of a geometric Brownian motion of the asset price St , dS = µ dt + σ dW.

Acta Applicandae Math. 2 (1984) 139-158. [Bi79] P. Billingsley: Probability and Measure. John Wiley, New York (1979). 284 References [BV00] [BS73] [Blo86] [BP00] [Bo98] [BoM58] [BBG97] [BTT00] [Br91] [BrS77] [BrS02] [Br94] [BrD97] [BrG97] [BrG04] [BH98] [BuJ92] [CaMO97] [CaF95] [CaM99] [Cash84] [CDG00] G.I. Bischi, V. Valori: Nonlinear effects in a discrete-time dynamic model of a stock market. Chaos, Solitons and Fractals 11 (2000) 21032121. F. Black, M. Scholes: The pricing of options and corporate liabilities. J. Political Economy 81 (1973) 637–659. E.C. Blomeyer: An analytic approximation for the American put price for options with dividends. J. Financial Quantitative Analysis 21 (1986) 229-233. J.-P. Bouchaud, M. Potters: Theory of Financial Risks. From Statistical Physics to Risk Management. Cambridge Univ.

pages: 489 words: 148,885

Accelerando by Stross, Charles


call centre, carbon-based life, cellular automata, cognitive dissonance, commoditize, Conway's Game of Life, dark matter, dumpster diving, Extropian, finite state, Flynn Effect, glass ceiling, gravity well, John von Neumann, knapsack problem, Kuiper Belt, Magellanic Cloud, mandelbrot fractal, market bubble, means of production, packet switching, performance metric, phenotype, planetary scale, Pluto: dwarf planet, reversible computing, Richard Stallman, SETI@home, Silicon Valley, Singularitarianism, slashdot, South China Sea, stem cell, technological singularity, telepresence, The Chicago School, theory of mind, Turing complete, Turing machine, Turing test, upwardly mobile, Vernor Vinge, Von Neumann architecture, web of trust, Y2K, zero-sum game

Amber has a nasty feeling that she's running in a compatibility sandbox here – there are signs that her access to the simulation system's control interface is very much via proxy – but at least she's got it. "Wow! Back in the real world at last!" She can hardly contain her excitement, even forgetting to be pissed at Sadeq for thinking she was just an actor in his Cartesian theatre's performance of Puritan Hell. "Look! It's the DMZ!" They're standing on a grassy knoll overlooking a gleaming Mediterranean city. It snoozes beneath a Mandelbrot-fuzzy not-sun that hangs at the center of a hyperbolic landscape, which dwindles into a blue yonder that seems incomprehensibly distant. Circular baby-blue wells open in the walls of the world at regular intervals, connecting to other parts of the manifold. "How big is it, ghost? In planetary simulation-equivalents." "This demilitarized zone is an embedded reality, funneling all transfers between the local star system's router and the civilization that built it.

Another fourteen months and the larger part of the cumulative conscious processing power of the human species will be arriving in silicon. And the first meat the new AIs get to know will be the uploaded lobsters. Manfred stumbles back to his hotel, bone-weary and jet-lagged; his glasses are still jerking, slashdotted to hell and back by geeks piggybacking on his call to dismantle the moon. They stutter quiet suggestions at his peripheral vision. Fractal cloud-witches ghost across the face of the moon as the last huge Airbuses of the night rumble past overhead. Manfred's skin crawls, grime embedded in his clothing from three days of continuous wear. Back in his room, the Aineko mewls for attention and strops her head against his ankle. She's a late-model Sony, thoroughly upgradeable: Manfred's been working on her in his spare minutes, using an open source development kit to extend her suite of neural networks.

"Their own fault; If they hadn't participated in antibiotic abuse they wouldn't be in the isolation ward," harrumphs a twentysomething with mutton-chops and the manner of a precocious paterfamilias. He raps his walking stick on the pavement for punctuation, and they pause for a flock of cyclists and a rickshaw before they cross the road onto the Meadows. "Degenerate medication compliance, degenerate immune systems." Manfred pauses to survey the grass, brain spinning as he ponders the fractal dimensionality of leaves. Then he lurches after them, nearly getting himself run down by a flywheel-powered tourist bus. Club. His feet hit the pavement, cross it, thud down onto three billion years of vegetative evolution. Something about those people. He feels a weird yearning, a tropism for information. It's almost all that's left of him – his voracious will to know. The tall, dark-haired woman hitches up her long skirts to keep them out of the mud. he sees a flash of iridescent petticoats that ripple like oil on water, worn over old-fashioned combat boots.

pages: 239 words: 68,598

The Vanishing Face of Gaia: A Final Warning by James E. Lovelock


Ada Lovelace, butterfly effect, carbon footprint, Clapham omnibus, cognitive dissonance, continuous integration, David Attenborough, decarbonisation, discovery of DNA, Edward Lorenz: Chaos theory, Henri Poincaré, Intergovernmental Panel on Climate Change (IPCC), mandelbrot fractal, mass immigration, megacity, Northern Rock, oil shale / tar sands, phenotype, Pierre-Simon Laplace, planetary scale, short selling, Stewart Brand, University of East Anglia

May found that computer models of population growth showed similar chaotic behaviour, especially in biological systems containing more than two species; these discoveries stirred great interest among mathematicians and scientists in the nature of deterministic chaos. Practical applications in communications and to new art forms have emerged, for example those stunning illustrations of fractal mathematics such as the Mandelbrot set. It was so human and apparently understandable that neither of these eminent scientists made much of the fact that the appearance of chaos suggested that something might be wrong with their hypotheses about the world. Lorenz and May were both looking at the Earth system from within separated scientific disciplines that took cause‐and‐effect determinism for granted. Yet if instead we look at climate and population growth as a single tightly coupled system we find the combined model is resilient to perturbation and makes credible predictions.

The insights from the numerical analysis of fluid dynamics by Edward Lorenz and of population biology by Robert May revealed what is called ‘deterministic chaos’. Systems like the weather, the motion of more than two astronomical bodies linked by gravitation, or more than two species in competition, are exceedingly sensitive to the initial conditions of their origin, and they evolve in a wholly unpredictable manner. The study of these systems is a rich and colourful new field of science enlivened by the visual brilliance of the strange images of fractal geometry. It is important to note that efficient dynamic mechanical systems, such as the autopilot of an aircraft, are essentially free of chaotic behaviour, and the same is true of healthy living organisms. Life can opportunistically employ chaos, but it is not a characteristic part of its normal function. CONSILIENCE The most distinguished evolutionary biologist E. O. Wilson, when writing on the incompatibility of twentieth‐century science and religion, was mindful of the unconscious need in most of us for something transcendental, something more than could come from cold analysis.

pages: 696 words: 143,736

The Age of Spiritual Machines: When Computers Exceed Human Intelligence by Ray Kurzweil


Ada Lovelace, Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, Any sufficiently advanced technology is indistinguishable from magic, Buckminster Fuller, call centre, cellular automata, combinatorial explosion, complexity theory, computer age, computer vision, cosmological constant, cosmological principle, Danny Hillis, double helix, Douglas Hofstadter, Everything should be made as simple as possible, first square of the chessboard / second half of the chessboard, fudge factor, George Gilder, Gödel, Escher, Bach, I think there is a world market for maybe five computers, information retrieval, invention of movable type, Isaac Newton, iterative process, Jacquard loom, Jacquard loom, John Markoff, John von Neumann, Lao Tzu, Law of Accelerating Returns, mandelbrot fractal, Marshall McLuhan, Menlo Park, natural language processing, Norbert Wiener, optical character recognition, ought to be enough for anybody, pattern recognition, phenotype, Ralph Waldo Emerson, Ray Kurzweil, Richard Feynman, Richard Feynman, Robert Metcalfe, Schrödinger's Cat, Search for Extraterrestrial Intelligence, self-driving car, Silicon Valley, speech recognition, Steven Pinker, Stewart Brand, stochastic process, technological singularity, Ted Kaczynski, telepresence, the medium is the message, There's no reason for any individual to have a computer in his home - Ken Olsen, traveling salesman, Turing machine, Turing test, Whole Earth Review, Y2K

Magnenat-Thalmann, Nadia and Daniel Thalmann. Computer Animation: Theory and Practice. Tokyo: Springer-Verlag, 1985. Malcolm, Norman. Ludwig Wittgenstein: A Memoir, with a Biographical Sketch by Georg Henrik Von Wright. Oxford: Oxford University Press, 1958. Mamdani, E. H. and B. R. Gaines. Fuzzy Reasoning and Its Applications. London: Academic Press, 1981. Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W H. Freeman, 1988. ─. Fractals: Form, Chance, and Dimension. San Francisco: W H. Freeman, 1977. Mander, Jerry. In the Absence of the Sacred: The Failure of Technology and the Survival of the Indian Nations. San Francisco: Sierra Club Books, 1992. Margulis, Lynn and Dorion Sagan. Microcosmos: Four Billion Years of Evolution from Our Microbial Ancestors. New York: Summit Books, 1986.

Los Altos, CA: Morgan Kaufmann, 1985. Nocera, Joseph. A Piece of the Action: How the Middle Class Joined the Money Class. New York: Simon and Schuster, 1994. Norretranders, Tor. The User Illusion: Cutting Consciousness Down to Size. New York: Viking, 1998. O’Keefe, Bernard J. Nuclear Hostages. Boston: Houghton-Mifflin Company, 1983. Oakley, D. A., ed. Brain and Mind. London and New York: Methuen, 1985. Oliver, Dick. FractalVision: Put Fractals to Work for You. Carmel, IN: Sams Publishing, 1992. Ornstein, Robert. The Evolution of Consciousness: Of Darwin, Freud, and Cranial Fire; the Origins of the Way We Think. New York: Prentice-Hall Press, 1991. ─. The Mind Fie!d. London: Octagon Press, 1976. ─. Multimind: A New Way of Looking at Human Behavior. Boston: Houghton-Mifflin, 1986. ─. On the Experience of Time. London: Penguin Books, 1969

Beyond Numeracy: Ruminations of a Number Man. New York: Alfred A. Knopf, 1991. Pavlov, I. P Conditioned Reflexes. London: Oxford University Press, 1927. Peat, F. David. Artificial Intelligence: How Machines Think. New York: Baen Enterprises, 1985. ________. Synchronicity: The Bridge Between Matter and Mind. Toronto: Bantam Books, 1987. Peitgen, H. O., D. Saupe, et al. The Science of Fractal Images. New York: Springer-Verlag, 1988. Peitgen, H. O. and P. H. Richter. The Beauty of Fractals: Images of Complex Dynamical Systems. Berlin: Springer-Verlag, 1986. Penfield, W The Mystery of the Mind. Princeton, NJ: Princeton University Press, 1975. Penrose, R. and C. J. Isham, eds. Quantum Concepts in Space and Time. Oxford: Oxford University Press: 1986. Penrose, Roger. The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics.

pages: 345 words: 86,394

Frequently Asked Questions in Quantitative Finance by Paul Wilmott


Albert Einstein, asset allocation, beat the dealer, Black-Scholes formula, Brownian motion, butterfly effect, capital asset pricing model, collateralized debt obligation, Credit Default Swap, credit default swaps / collateralized debt obligations, delta neutral, discrete time, diversified portfolio, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, fixed income, fudge factor, implied volatility, incomplete markets, interest rate derivative, interest rate swap, iterative process, London Interbank Offered Rate, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, market bubble, martingale, Myron Scholes, Norbert Wiener, Paul Samuelson, quantitative trading / quantitative finance, random walk, regulatory arbitrage, risk/return, Sharpe ratio, statistical arbitrage, statistical model, stochastic process, stochastic volatility, transaction costs, urban planning, value at risk, volatility arbitrage, volatility smile, Wiener process, yield curve, zero-coupon bond

• As long as the market maker has a positive expectation for each trade, although with some model risk, having a large number of positions he will reduce exposure overall by diversification. This is more like an actuarial approach to model risk. • If neither of the above is possible then he could widen his bid-ask spreads. He will then only trade with those people who have significantly different market views from him. References and Further Reading Mandelbrot, B & Hudson, R 2004 The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward. Profile Books How Robust is the Black-Scholes Model? Short Answer Very robust. You can drop quite a few of the assumptions underpinning Black-Scholes and it won’t fall over. Example Transaction costs? Simply adjust volatility. Time-dependent volatility? Use root-mean-square-average volatility instead. Interest rate derivatives?

If Wt is the BM at time t then for every t, τ ≥ 0, Wt+τ − Wt is independent of {Wu : 0 ≤ u ≤ t}, and has a normal distribution with zero mean and variance τ. The important properties of BM are as follows. • Finiteness: the scaling of the variance with the time step is crucial to BM remaining finite. • Continuity: the paths are continuous, there are no discontinuities. However, the path is fractal, and not differentiable anywhere. • Markov: the conditional distribution of Wt given information up until τ < t depends only on Wτ . • Martingale: given information up until τ < t the conditional expectation of Wt is Wτ. • Quadratic variation: if we divide up the time 0 to t in a partition with n + 1 partition points ti = it/n then • Normality: Over finite time increments ti−1 to ti, is normally distributed with mean zero and variance ti − ti−1.

After the second iteration you add an area that is number of sides multiplied by area of a single small triangle which is one ninth of the previously added triangle. If we use An to be the area after n iterations (when multiplied by the area of initial triangle) then So The final calculation exploits the binomial expansion. This is the famous Koch snowflake, first described in 1904, and is an example of a fractal. The doors There are one hundred closed doors in a corridor. The first person who walks along the corridor opens all of the doors. The second person changes the current state of every second door starting from the second door by opening closed doors and closing open doors. The third person who comes along changes the current state of every third door starting from the third door. This continues until the 100th person.

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The Death of Money: The Coming Collapse of the International Monetary System by James Rickards


Affordable Care Act / Obamacare, Asian financial crisis, asset allocation, Ayatollah Khomeini, bank run, banking crisis, Ben Bernanke: helicopter money, bitcoin, Black Swan, Bretton Woods, BRICs, business climate, capital controls, Carmen Reinhart, central bank independence, centre right, collateralized debt obligation, collective bargaining, complexity theory, computer age, credit crunch, currency peg, David Graeber, debt deflation, Deng Xiaoping, diversification, Edward Snowden, eurozone crisis, fiat currency, financial innovation, financial intermediation, financial repression, fixed income, Flash crash, floating exchange rates, forward guidance, G4S, George Akerlof, global reserve currency, global supply chain, Growth in a Time of Debt, income inequality, inflation targeting, information asymmetry, invisible hand, jitney, John Meriwether, Kenneth Rogoff, labor-force participation, labour mobility, Lao Tzu, liquidationism / Banker’s doctrine / the Treasury view, liquidity trap, Long Term Capital Management, mandelbrot fractal, margin call, market bubble, market clearing, market design, money market fund, money: store of value / unit of account / medium of exchange, mutually assured destruction, obamacare, offshore financial centre, oil shale / tar sands, open economy, Plutocrats, plutocrats, Ponzi scheme, price stability, quantitative easing, RAND corporation, reserve currency, risk-adjusted returns, Rod Stewart played at Stephen Schwarzman birthday party, Ronald Reagan, Satoshi Nakamoto, Silicon Valley, Silicon Valley startup, Skype, sovereign wealth fund, special drawing rights, Stuxnet, The Market for Lemons, Thomas Kuhn: the structure of scientific revolutions, Thomas L Friedman, too big to fail, trade route, uranium enrichment, Washington Consensus, working-age population, yield curve

No Ancient Wisdom, No Followers. Westport, Conn.: Prospecta Press, 2012. Mackay, Charles. Extraordinary Popular Delusions and the Madness of Crowds. New York: Farrar, Straus and Giroux, 1932. McKinnon, Ronald I. The Unloved Dollar Standard: From Bretton Woods to the Rise of China. Oxford: Oxford University Press, 2013. Mandelbrot, Benoit. The Fractal Geometry of Nature. New York: W. H. Freeman, 1983. Mandelbrot, Benoit, and Richard L. Hudson. The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. New York: Basic Books, 2004. Martines, Lauro. Furies: War in Europe, 1450–1700. New York: Bloomsbury, 2013. Marx, Karl. Selected Writings. Edited by David McLellan. Oxford: Oxford University Press, 1977. Mead, Walter Russell. God and Gold: Britain, America, and the Making of the Modern World.

pages: 626 words: 181,434

I Am a Strange Loop by Douglas R. Hofstadter


Albert Einstein, Andrew Wiles, Benoit Mandelbrot, Brownian motion, double helix, Douglas Hofstadter, Georg Cantor, Gödel, Escher, Bach, Isaac Newton, James Watt: steam engine, John Conway, John von Neumann, mandelbrot fractal, pattern recognition, Paul Erdős, place-making, probability theory / Blaise Pascal / Pierre de Fermat, publish or perish, random walk, Ronald Reagan, self-driving car, Silicon Valley, telepresence, Turing machine

This was the case because the most unpredictable of the visual phenomena always seemed to happen right in the vicinity of that central point where the infinite regress converges down to a magical dot. 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. The key question is this: If you, playing the role of microphone and loudspeaker (or camera and TV), do this over and over again, will the z values you get grow unboundedly, sailing off into the wild blue (or wild yellow or wild red) yonder, or will they instead home in on a finite value?

Machine Z machines: with beliefs about free will; confused; conscious; creative; dedicated; downloading of; emulating other machines; with linguistic capacity; as necessarily unconscious; with opinions; reading and interpreting description of own structure; with souls; universal; who think vs. that think Machines Who Think (McCorduck) Macintosh, emulating Alienware machine MacLaine, Shirley macroscopic boundaries, irrelevant to particles macroscopic forces as patterns Madurodam Magellan, Ferdinand magic genie for all mathematical questions magic square, ill-definedness of the notion magical realism magical thinking connected with “I”-ness “magical” vs. “ordinary” entities magnanimity; etymology of the word “Mahatma”, etymology of “main brain” of a given soul Malagasy language, presumed opacity of Mallory, George Malraux, André mammals as dividing line for food Mandelbrot, Benoit Mann, Lois Mantle, Mickey mapping: of colors to color sensations; at core of life; giving rise to meaning; of PM patterns into the world of numbers “marbelous”, too much for words marble, illusory, in envelope box; see also Epi Margolin, Janet marital bond, tightness of Marot, Clément marriage: of Carol and Doug; as soul merger; as third patient in counseling for a couple Married People: Staying Together in the Age of Divorce (Klagsbrun) Mars, teleportation to Martin, Mary Martin, Richard M.

When we symbol-possessing humans watch a video feedback system, we naturally pay attention to the eye-catching shapes on the screen and are seduced into giving them fanciful labels like “helical corridor” or “galaxy”, but still we know that ultimately they consist of nothing but pixels, and that whatever patterns appear before our eyes do so thanks solely to the local logic of pixels. This simple and clear realization strips those fancy fractalic gestalts of any apparent life or autonomy of their own. We are not tempted to attribute desires or hopes, let alone consciousness, to the screen’s swirly shapes — no more than we are tempted to perceive fluffy cotton-balls in the sky as renditions of an artist’s profile or the stoning of a martyr. And yet when it comes to perceiving ourselves, we tell a different story. Things are far murkier when we speak of ourselves than when we speak of video feedback, because we have no direct access to any analogue, inside our brains, to pixels and their local logic.

pages: 232

Planet of Slums by Mike Davis


barriers to entry, Branko Milanovic, Bretton Woods, British Empire, Brownian motion, centre right, clean water, conceptual framework, crony capitalism, declining real wages, deindustrialization, Deng Xiaoping, edge city, European colonialism, failed state, Gini coefficient, Hernando de Soto, housing crisis, illegal immigration, income inequality, informal economy, Intergovernmental Panel on Climate Change (IPCC), Internet Archive, jitney, jobless men, Kibera, labor-force participation, land reform, land tenure, liberation theology, low-wage service sector, mandelbrot fractal, market bubble, megacity, microcredit, New Urbanism, Pearl River Delta, Ponzi scheme, RAND corporation, rent control, structural adjustment programs, surplus humans, upwardly mobile, urban planning, urban renewal, War on Poverty, Washington Consensus, working poor

"Partly these fires, " Hans Schenk writes, "are said to be organized by slum leaders who can cash (part of) the government compensation money; partly by some political party-affiliated gangs to clear 'unwelcome' categories of the urban poor; partly by private landowners who want their land cleared in an easy way from (illegal) squatters and have it 'developed.'"24 Pathologies of Urban Form If natural hazards are magnified by urban poverty, new and entirely artificial hazards are created by poverty's interactions with toxic industries, anarchic traffic, and collapsing infrastructures. The chaotic form of so many Third World cities — "urban mandelbrots," according to urban theorist Matthew Gandy — annuls much of the environmental 24 Hans Schenk, "Living in Bangalore's Slums," in Schenk (ed.), Living in India's Slums: A. Case Study of Bangalore, Delhi 2001, p. 34. efficiency of city life and breeds the small disasters that constantly terrorize metropolises like Mexico City, Cairo, Dhaka, and Lagos. ("Lagos," explains Gandy, "does not really exist as a city in a conventional sense: its boundaries are unclear; many of its constituent elements appear to function independently of one another....") 23 All the classical principles of urban planning, including the preservation of open space and the separation of noxious land uses from residences, are stood on their heads in poor cities.

example, in 1992 had an estimated 6.6 million low-income people living contiguously in 348 square kilometers of informal housing.28 Most of the poor in Lima, likewise, live in three great peripheral corns radiating from the central city; such huge spatial concentrations of urban poverty are also common in Africa and the Middle East. In South Asia, on the other hand, the urban poor tend to live in a much larger number of distinct slums more widely dispersed throughout the urban fabric in patterns with an almost fractal complexity. In Kolkata, for instance, thousands of thika bustees — nine hutments of five huts each, with 45square-meter rooms shared, on average, by an incredible 13.4 people — are intermixed with a variety of other residential statuses and landuses 29 In Dhaka, it probably makes more sense to consider the nonslum areas as enclaves in an overwhelming matrix of extreme poverty. Although some slums have long histories — Rio de Janeiro's first favela, Morro de Providencia, was founded in the 1880s — most megaslums have grown up since the 1960s.

pages: 240 words: 73,209

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


Albert Einstein, Atul Gawande, Benoit Mandelbrot, big-box store, Black Swan, Checklist Manifesto, Clayton Christensen, Daniel Kahneman / Amos Tversky, Exxon Valdez, Gordon Gekko, housing crisis, information asymmetry, Isaac Newton, Kenneth Arrow, Long Term Capital Management, Mahatma Gandhi, mandelbrot fractal, NetJets, pattern recognition, pre–internet, random walk, Ronald Reagan, South Sea Bubble, Steve Jobs, winner-take-all economy, young professional, zero-sum game

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? Or, A Good Hard Look at Wall Street by Fred Schwed Your Money and Your Brain: How the New Science of Neuroeconomics Can Help Make You Rich by Jason Zweig Literature 100 Years of Solitude by Gabriel García Márquez Hamlet by William Shakespeare Jonathan Livingston Seagull by Richard Bach Oliver Twist by Charles Dickens Zen and the Art of Motorcycle Maintenance: An Inquiry into Values by Robert Pirsig Miscellaneous Autobiography: The Story of My Experiments with the Truth by Mahatma Gandhi City Police by Jonathan Rubinstein Endurance: Shackleton’s Incredible Voyage by Alfred Lansing Long Walk to Freedom: The Autobiography of Nelson Mandela by Nelson Mandela Metaphors We Live By by George Lakoff and Mark Johnson Reagan: A Life in Letters by Ronald Reagan The Autobiography of Benjamin Franklin by Benjamin Franklin The Checklist Manifesto: How to Get Things Right by Atul Gawande The Hero with a Thousand Faces by Joseph Campbell The New British Constitution by Vernon Bogdanor The Power of Myth by Joseph Campbell with Bill Moyers Vor 1914: Erinnerungen an Frankfurt geschrieben in Israel by Selmar Spier Walden: or, Life in the Woods by Henry David Thoreau Why America Is Not a New Rome by Vaclav Smil Philosophy and Theology A Theory of Justice by John Rawls Anarchy, the State, and Utopia by Robert Nozick Destination Torah: Reflections on the Weekly Torah Readings by Isaac Sassoon Halakhic Man by Joseph Soloveitchik Letters from a Stoic by Lucius Annaeus Seneca Man’s Search for Meaning by Viktor Frankl Meditations by Marcus Aurelius Pirke Avot: A Modern Commentary on Jewish Ethics by Leonard Kravits and Kerry Olitzky Plato, not Prozac!

pages: 349 words: 134,041

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


accounting loophole / creative accounting, Albert Einstein, Asian financial crisis, asset-backed security, beat the dealer, Black Swan, Black-Scholes formula, Bretton Woods, BRICs, Brownian motion, business process, buy low sell high, call centre, capital asset pricing model, collateralized debt obligation, commoditize, complexity theory, computerized trading, corporate governance, corporate raider, Credit Default Swap, credit default swaps / collateralized debt obligations, cuban missile crisis, currency peg, disintermediation, diversification, diversified portfolio, Edward Thorp, Eugene Fama: efficient market hypothesis, Everything should be made as simple as possible, financial innovation, fixed income, Haight Ashbury, high net worth, implied volatility, index arbitrage, index card, index fund, interest rate derivative, interest rate swap, Isaac Newton, job satisfaction, John Meriwether, locking in a profit, Long Term Capital Management, mandelbrot fractal, margin call, market bubble, Marshall McLuhan, mass affluent, mega-rich, merger arbitrage, Mexican peso crisis / tequila crisis, money market fund, moral hazard, mutually assured destruction, Myron Scholes, new economy, New Journalism, Nick Leeson, offshore financial centre, oil shock, Parkinson's law, placebo effect, Ponzi scheme, purchasing power parity, quantitative trading / quantitative finance, random walk, regulatory arbitrage, Right to Buy, risk-adjusted returns, risk/return, Satyajit Das, shareholder value, short selling, South Sea Bubble, statistical model, technology bubble, the medium is the message, the new new thing, time value of money, too big to fail, transaction costs, value at risk, Vanguard fund, volatility smile, yield curve, Yogi Berra, zero-coupon bond

Fischer Black thought traders should have a ‘story’ about why they traded in the way they did. Making money didn’t prove anything, it could just be a lucky accident. A good story did not guarantee success, a bad one just meant that the trader was riding his luck, shooting craps. Quants increasingly ‘mine’ vast quantities of data to ‘prove’ their models. You can substantiate anything given enough data. There is ‘chaos’ theory – the world of fractals, the eponymous Mandelbrot set. In popular science, chaos theory is portrayed as the relationship between the perfect storm and the wing beats of a butterfly in the Amazon jungle. In truth, it is a form of non-Euclidean geometry (the stuff you learn at school). It is used to model complex phenomenon including, hilariously, financial markets. The irony of trying to model chaos, the finding of order in complete disorder, is lost on most quants.

(You need to show larger losses than the last rogue trader the firm employed.) Selection criteria • You will be able to demonstrate a detailed knowledge of financial markets and trading techniques. (You should wax lyrical about obscure markets (the Zambian Kwatcho and L Key responsibilities DAS_Z01.QXP 8/11/06 2:10 PM Page 315 Epilogue 315 Table E.1 Continued • • • • • Islamic finance techniques) and complex mathematics (field theory; neural networks; fractals). Everybody will think you are a genius or a fool but will be unsure of which.) You will be able to demonstrate detailed knowledge of derivatives, including exotic and non-standard structures. (Everybody knows that derivatives allow highly leveraged positions that are impossible to understand or value accurately.) No minimum formal educational qualifications or direct previous experience in a similar role is necessary.

pages: 488 words: 148,340

Aurora by Kim Stanley Robinson


back-to-the-land, cognitive bias, cognitive dissonance, dark matter, epigenetics, gravity well, mandelbrot fractal, microbiome, traveling salesman, Turing test

Still, these were highly perturbed, one might even say chaotic gravity eddies, and though their pull was very slight, and we seldom flew through one anyway, they still needed to be attended to in the algorithms, and used or compensated for as the case might be. Jupiter: we came in just past the molten yellow sulfuric black-spotted ball of Io, aimed for a periapsis that was just slightly inside the uppermost gas clouds of the great banded gas giant, all tans and ochres and burnt siennas, with the wind-sheared border between each equatorial band an unctuous swirl of Mandelbrot paisleys, looking much more viscous than they really were, being fairly diffuse gases up there at the top of the atmosphere, but sharply delineated by densities and gas contents, apparently, because no matter how close we came the impression remained. We came in around the equator, above a little dimple that was apparently the remnant of the Great Red Spot, which had collapsed in the years 2802–09.

There were limits on how many people could leave the shelters at once, so there was a scramble for spots on the schedule during this slack time, because at some point in the early afternoon of the daymonth, the onshore wind would begin, a hard flow of air barreling in off the sea into the interior of Greenland, as the land got hotter than the ocean and its air rose and vacated a space that cooler sea air rushed in to fill, the wind arriving in puffs and faltering breezes, then in a steady gentle push, which strengthened through the afternoon of the daymonth until sunset. This was generally the time of strongest onshore winds, although that varied of course, as storm systems swirled around Aurora in the usual fractal nautiloid motions that occur when gases move around the exterior of a rotating sphere. Although Aurora’s day was also its month, it was still rotating once in that daymonth, and that slow rotation caused the air in the atmosphere to drag a little in relation to both hydrosphere and lithosphere, creating winds that curled and mixed to create the usual trades, polar swirls, and so on. So: almost always windy.

Outside that… if ship were flying in intergalactic space, the medium would presumably be that much more diffuse. Visible around any ship in the intergalactic medium would be galaxies like stars. They would cluster irregularly, as stars cluster within a galaxy. The greater structure of galactic diffusion would become visible; clouds of galaxies like gas clouds, then the Great Wall, then also emptier bubbles where few or no galaxies reside. The universe is fractal; and even when flying inside a galaxy, this vision of galaxies clustering around us out to the universal horizon is available, using certain filters. Granular vision in different registers. Something like a septillion stars in the observable universe, we calculate, but also there may be as many universes as there are stars in this universe, or atoms. An itch. A faint hissing. A waft of smoke on a breeze.

Adaptive Markets: Financial Evolution at the Speed of Thought by Andrew W. Lo

Albert Einstein, Alfred Russel Wallace, algorithmic trading, Andrei Shleifer, Arthur Eddington, Asian financial crisis, asset allocation, asset-backed security, backtesting, bank run, barriers to entry, Berlin Wall, Bernie Madoff, bitcoin, Bonfire of the Vanities, bonus culture, break the buck, Brownian motion, business process, butterfly effect, capital asset pricing model, Captain Sullenberger Hudson, Carmen Reinhart, Chance favours the prepared mind, collapse of Lehman Brothers, collateralized debt obligation, commoditize, computerized trading, corporate governance, creative destruction, Credit Default Swap, credit default swaps / collateralized debt obligations, cryptocurrency, Daniel Kahneman / Amos Tversky, delayed gratification, Diane Coyle, diversification, diversified portfolio, double helix, easy for humans, difficult for computers, Ernest Rutherford, Eugene Fama: efficient market hypothesis, experimental economics, experimental subject, Fall of the Berlin Wall, financial deregulation, financial innovation, financial intermediation, fixed income, Flash crash, Fractional reserve banking, framing effect, Gordon Gekko, greed is good, Hans Rosling, Henri Poincaré, high net worth, housing crisis, incomplete markets, index fund, interest rate derivative, invention of the telegraph, Isaac Newton, James Watt: steam engine, job satisfaction, John Maynard Keynes: Economic Possibilities for our Grandchildren, John Meriwether, Joseph Schumpeter, Kenneth Rogoff, London Interbank Offered Rate, Long Term Capital Management, loss aversion, Louis Pasteur, mandelbrot fractal, margin call, Mark Zuckerberg, market fundamentalism, martingale, merger arbitrage, meta analysis, meta-analysis, Milgram experiment, money market fund, moral hazard, Myron Scholes, Nick Leeson, old-boy network, out of africa, p-value, paper trading, passive investing, Paul Lévy, Paul Samuelson, Ponzi scheme, predatory finance, prediction markets, price discovery process, profit maximization, profit motive, quantitative hedge fund, quantitative trading / quantitative finance, RAND corporation, random walk, randomized controlled trial, Renaissance Technologies, Richard Feynman, Richard Feynman, Richard Feynman: Challenger O-ring, risk tolerance, Robert Shiller, Robert Shiller, short selling, sovereign wealth fund, statistical arbitrage, Steven Pinker, stochastic process, survivorship bias, The Great Moderation, the scientific method, The Wealth of Nations by Adam Smith, The Wisdom of Crowds, theory of mind, Thomas Malthus, Thorstein Veblen, Tobin tax, too big to fail, transaction costs, Triangle Shirtwaist Factory, ultimatum game, Upton Sinclair, US Airways Flight 1549, Walter Mischel, Watson beat the top human players on Jeopardy!, WikiLeaks, Yogi Berra, zero-sum game

Journal of Financial Markets 6: 163–197. Mallaby, Sebastian. 2010. More Money than God: Hedge Funds and the Making of a New Elite. New York: Penguin Books. Maloney, Michael T., and J. Harold Mulherin. 2003. “The Complexity of Price Discovery in an Efficient Market: The Stock Market Reaction to the Challenger Crash.” Journal of Corporate Finance 9: 453– 479. References • 453 Mandelbrot, Benoit B. 1982. The Fractal Geometry of Nature. San Francisco: W. H. Freeman. Markowitz, Harry. 1952. “Portfolio Selection.” Journal of Finance 7: 77–91. Marshall, Alfred. 2009. Principles of Economics: Unabridged Eighth Edition. New York: Cosimo. May, Robert M., Simon A. Levin, and George Sugihara. 2008. “Ecology for Bankers.” Nature 451: 893–895. Maynard Smith, John. 1975. “Survival by Suicide.” New Scientist 67:496– 497. –––. 1982.

Th is unusual term derives from an eighteenth-century gambling strategy in which bettors would double their stakes to cover their previous losses (which is definitely not a good idea, but often too tempting to resist for reasons we’ll get to in chapters 3 and 4). 13. See Bass (1985) and the fictionalized account of Mezrich (2002) for examples. 14. Einstein (1905). 15. In his evaluation of his Ph.D. student’s rather unorthodox thesis, Poincaré highlighted the curious connection between science and financial economics (Mandelbrot [1982, 395]): The manner in which the candidate obtains the law of Gauss is most original, and all the more interesting as the same reasoning might, with a few changes, be extended to the theory of errors. He develops this in a chapter which might at fi rst seem strange, for he titles it “Radiation of Probability.” In effect, the author resorts to a comparison with the analytical theory of the propagation of heat.

International Monetary Fund Working Paper WP/11/193. Ezekiel, Mordecai. 1938. “The Cobweb Theorem.” Quarterly Journal of Economics 52: 255–280. Fagnan, David, Jose Maria Fernandez, Andrew W. Lo, and Roger M. Stein. 2013. “Can Financial Engineering Cure Cancer?” American Economic Review 103: 406– 411. Falk, Dean. 1990. “Brain Evolution in Homo: The “Radiator” Theory.” Behavioral and Brain Sciences 13: 333–344. Fama, Eugene. 1963. “Mandelbrot and the Stable Paretian Hypothesis.” Journal of Business 36: 420–29. –––. 1965a. “The Behavior of Stock Market Prices.” Journal of Business 38: 34–105. –––. 1965b. “Random Walks in Stock Market Prices.” Financial Analysts Journal 21: 55–59. –––. 1970. “Efficient Capital Markets: A Review of Theory and Empirical Work.” Journal of Finance 25: 383– 417. Fama, Eugene, Lawrence Fisher, Michael Jensen, and Richard Roll. 1969.

pages: 379 words: 109,612

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


A Declaration of the Independence of Cyberspace, Albert Einstein, AltaVista, Amazon Mechanical Turk, Asperger Syndrome, availability heuristic, Benoit Mandelbrot, biofilm, Black Swan, British Empire, conceptual framework, corporate governance, Danny Hillis, Douglas Engelbart, Douglas Engelbart, Emanuel Derman, epigenetics, Flynn Effect, Frank Gehry, Google Earth, hive mind, Howard Rheingold, index card, information retrieval, Internet Archive, invention of writing, Jane Jacobs, Jaron Lanier, John Markoff, Kevin Kelly, lifelogging, lone genius, loss aversion, mandelbrot fractal, Marc Andreessen, Marshall McLuhan, Menlo Park, meta analysis, meta-analysis, New Journalism, Nicholas Carr, out of africa, Paul Samuelson, peer-to-peer, Ponzi scheme, pre–internet, Richard Feynman, Richard Feynman, Rodney Brooks, Ronald Reagan, Schrödinger's Cat, Search for Extraterrestrial Intelligence, SETI@home, Silicon Valley, Skype, slashdot, smart grid, social graph, social software, social web, Stephen Hawking, Steve Wozniak, Steven Pinker, Stewart Brand, Ted Nelson, telepresence, the medium is the message, the scientific method, The Wealth of Nations by Adam Smith, theory of mind, trade route, upwardly mobile, Vernor Vinge, Whole Earth Catalog, X Prize

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.

The Art of Computer Programming by Donald Ervin Knuth


Brownian motion, complexity theory, correlation coefficient, Donald Knuth, Eratosthenes, Georg Cantor, information retrieval, Isaac Newton, iterative process, John von Neumann, Louis Pasteur, mandelbrot fractal, Menlo Park, NP-complete, P = NP, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, RAND corporation, random walk, sorting algorithm, Turing machine, Y2K

Therefore if x and y are arbitrary reals and k > 1, the number Zk = ([16fcccJ + [16fcyjz)/16fc is in S + m + ni for some integers m and n. It can be shown that S + m + ni is bounded away from the origin when (m, n) / @,0). Consequently if \x\ and \y\ are fixed and k is sufficiently large, we have Zk G S, and limfc-^oo Zk = x + yi is in S. [B. Mandelbrot named S the "twindragon" because he noticed that it is essentially obtained by joining two "dragon curves" belly-to-belly; see his book Fractals: Form, Chance, and Dimension (San Francisco: Freeman, 1977), 313-314, where he also stated that the dimension of the boundary is 2 lg cc « 1.523627, where x = l + 2cc~2 s=: 1.69562. Other properties of the dragon curve are described in C. Davis and D. E. Knuth, J. Recr. 4.1 ANSWERS TO EXERCISES 607 Math. 3 A970), 66-81, 133-149.

Vittorio Griinwald proposed using the digits 0 and l/\/2 in odd-numbered positions, to avoid such a problem; but that actually spoils the whole system [see Commentari dell'Ateneo di Brescia A886), 43-54]. 206 ARITHMETIC +1-* Fig. 1. The fractal set 5 called the "twindragon." Another "binary" complex number system may be obtained by using the base i - 1, as suggested by W. Penney [JACM 12 A965), 247-248]: 4a4 i)a3 - 2ia2 a0 - | _i H In this system, only the digits 0 and 1 are needed. One way to demonstrate that every complex number has such a representation is to consider the interesting set S shown in Fig. 1; this set is, by definition, all points that can be written as Ylk>iQ<k(i — l)~fc5 for an infinite sequence ai, a2, a^, ... of zeros and ones. It is also known as the "twindragon fractal" [see M. F. Barnsley, Fractals Everywhere, second edition (Academic Press, 1993), 306, 310]. Figure 1 shows that S can be decomposed into 256 pieces congruent to j^S.

MacLaren, Malcolm Donald, 33, 47, 128, 551, 585. MacMahon, Percy Alexander, 609. MacMillan, Donald B., 226. MacPherson, Robert Duncan, 114. 750 INDEX AND GLOSSARY MacSorley, Olin Lowe, 280. Maeder, Roman Erich, 627, 635. Mahler, Kurt, 180. measure, 683. Makarov, Oleg Mikhailovich (MaicapoB, Ojier MnxafijioBHi), 700, 714. Mallows, Colin Lingwood, 74. . Manasse, Mark Steven, 403. Manchester University Computer, 192. Mandelbrot, Benoit Baruch, 606. Mangoldt, Hans Carl Friedrich von, 663. function, 371, 376. MANIAC III computer, 242. Mansour, Yishay (Ti^a >W>), 316. Mantel, Willem, 552. Mantissa, 214, see Fraction part. Marcziriski, R. W., 205. Mariage, Aime, 201. Mark I computer (Ferranti), 3. Mark II Calculator (Harvard), 225. Marsaglia, George, 3, 23, 29, 33, 40, 47, 62, 71, 75, 78, 108, 114-115, 119, 122, 123, 128, 133-135, 179, 544, 546-547, 549, 551, 565, 588.

pages: 661 words: 169,298

Coming of Age in the Milky Way by Timothy Ferris


Albert Einstein, Albert Michelson, Alfred Russel Wallace, anthropic principle, Arthur Eddington, Atahualpa, Cepheid variable, Chance favours the prepared mind, Commentariolus, cosmic abundance, cosmic microwave background, cosmological constant, cosmological principle, dark matter, delayed gratification, Edmond Halley, Eratosthenes, Ernest Rutherford, Gary Taubes, Harlow Shapley and Heber Curtis, Harvard Computers: women astronomers, Henri Poincaré, invention of writing, Isaac Newton, John Harrison: Longitude, Karl Jansky, Lao Tzu, Louis Pasteur, Magellanic Cloud, mandelbrot fractal, Menlo Park, Murray Gell-Mann, music of the spheres, planetary scale, retrograde motion, Richard Feynman, Richard Feynman, Search for Extraterrestrial Intelligence, Searching for Interstellar Communications, Solar eclipse in 1919, source of truth, Stephen Hawking, Thomas Kuhn: the structure of scientific revolutions, Thomas Malthus, Wilhelm Olbers

Chicago: University of Chicago Press, 1963. Mahaffey, J.P. Greek Life and Thought from the Age of Alexander to the Roman Conquest. London: Macmillan, 1887. Mainx, Felix. Foundations of Biology. Chicago: University of Chicago Press, 1955. Malthus, Thomas Robert. An Essay on the Principle of Population, ed. Philip Appleman. New York: Norton, 1976. Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: Freeman, 1983. Introduction to fractal geometry, by its founder. Manier, E. The Young Darwin and His Cultural Circle. Boston: Reidel, 1978. Manuel, Frank E. A Portrait of Isaac Newton. Washington, D.C.: New Republic, 1968. Psychological study. Marchant, James. Alfred Russel Wallace: Utters and Reminiscences. 2 vols. London: Cassel & Co., 1916. Marques, A.H. de Oliveira. History of Portugal.

Gödel’s theorem suggests that we never will—that a theory by its very nature requires for its verification the existence or contemplation of a larger reference frame. It is the boundary condition, then, that provides the essential distinction between mind and the universe: Thoughts and events are bounded, even if the totality is not.* And where did the boundaries come from? Quite possibly from the breaking of cosmic symmetries at the moment of genesis. We look out across a cosmic landscape riven by the fractal lines of broken symmetries, and draw from their patterns metaphors that aspire to be as creative, if not always quite as flawed, as the universe they purport to describe. (All metaphors are imperfect, said the poet Robert Frost, that is the beauty of them.) It may be, then, that the universe is comprehensible because it is defective—that because it forsook the perfection of nonbeing for the welter of being, it is possible for us to exist, and to perceive the jumbled, blemished reality, and to test it against the ghostly specter of the primordial symmetry thought to have preceded it.

pages: 1,079 words: 321,718

Surfaces and Essences by Douglas Hofstadter, Emmanuel Sander


affirmative action, Albert Einstein, Arthur Eddington, Benoit Mandelbrot, Brownian motion, Chance favours the prepared mind, cognitive dissonance, computer age, computer vision, dematerialisation, Donald Trump, Douglas Hofstadter, Ernest Rutherford, experimental subject, Flynn Effect, Georg Cantor, Gerolamo Cardano, Golden Gate Park, haute couture, haute cuisine, Henri Poincaré, Isaac Newton, l'esprit de l'escalier, Louis Pasteur, Mahatma Gandhi, mandelbrot fractal, Menlo Park, Norbert Wiener, place-making, Sapir-Whorf hypothesis, Silicon Valley, statistical model, Steve Jobs, Steve Wozniak, theory of mind, upwardly mobile, urban sprawl, yellow journalism, zero-sum game

.), 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.

Martin’s purchase belongs to, 189–190; of categorizations having opaque mechanisms, 511; of category distinctions never taught in schools, 126, 127; of children’s semantic approximations, 39–41; of chunked items in grocery stores, 92; of cognitive-dissonance reduction situations, 116; of colorful acts of categorization, 510; of common metaphorical uses of words, 62; of compound words in French and Italian, 89; of compound words with “air”, 86; of compound words with rival plurals, 87; of compound words with unnoticed components, 86; of computer concepts used in daily life, 402–404; of concepts at the core of a conceptual space, 79; of concepts close to golfer, 49, 50; of concepts modified by their “children”, 53–54; of concepts tacit in the concept hub, 52, 54; of concepts whose members have great reality to us, 132; of concepts with blurry boundaries, 60; of concepts with labels of ever-greater length, 111; of conceptual broadenings of catchy new concepts, 130; of conceptual-proximity slippage errors, 271–276; of conjunctions that name categories, 55, 70; of consequences of special relativity, 468; of containment situations in everyday life, 333; of conventional metaphorical usages, 232; of cousins of the word “and”, 72; of cousins of the word “but”, 30, 72, 74; of criteria for bird-ness, 55; of “dead” acronyms, 93; of decision-making situations, 330; of definitions of intelligence, 125; of definitions of multiplication, 412; of different ways of eating, 10; of diverse factors influencing categorization, 526; of diverse forms of caricature analogies, 320; of division word-problems concocted by students, 416; of division word-problems that give a larger answer, 418; of doctors who generalize known cures, 463–465; of dog breeds, 238; of English phrases used in French, 122; of entities belonging to rival categories, 191–192; of “equations” in advertisements, 409; of errors mediated by mutually reinforcing analogies, 277–278; of everyday analogies, 507; of everyday concepts of boundless richness, 5; of fairly low-frequency concepts, 81; of familiar concepts, 390; of families of metaphors, 63; of famous golfers of yore, 49; of fancy names used by professionals for familiar things, 421; of fancy technical concepts, 51; of fauxthenticity examples, 176–177, 178; of features of attics, 48–49; of features of offices, 47; of features of studies, 47–48; of four division problems involving photos, 422; of frame-blending Copycat analogies, 359; of French phrases used in English, 122; of French words for “pattern”, 81; of French words for “to get”, 80; of gearshift attributes, 344; of genericized brand names, 217; of generic-male usages, 193; of geniuses of yore who would be astounded by today’s commonplace knowledge, 130; of great commanders, 125; of highly variegated categories, 516; of historical precedents of the Vietnam War, 332; of household-item compound words, 88; of human needs engendering cultural activities, 314; of hypothetical contributions today of yesterday’s geniuses, 132; of hypothetical idioms for “spill the beans”, 96; of idiomatic phrases, 95; of ignored aspects of familiar things, 427; of impenetrable idioms, 105; of inferences made from category membership, 20, 21; of ingrained habit suddenly turns obsolete situations, 149; of items frequently used in division wordproblems, 419; of Jewish-mother jokes, 93; of Jewish-mother traits, 94; of labeled concepts, 20; of levels of abstraction in the speech chain, 25–26; of lexical blends, 260; of life lessons derived from Pac-Man, 303; of lists in this book, 569–570; of lovely spots in San Francisco, 296; of Mandarin verbs for playing instruments, 12; of marginal category-memberships, 56; of marginal members of the category bridge, 67; of marking in language, 193–194; of maternal traits, 34, 38; of meanings of the French word “ciel”, 375; of meanings of the word “band”, 3–4; of meccas, 220; of mechanical experiments, types of, 466–467; of medium-frequency concepts, 81; of members of the bark worse than bite category, 96; of members of the category unfortunate incidents caused by trying to avoid them, 524, 526; of members of the category very, 75; of metaphorical mothers, 38; of metaphorical usages of “to break”, 42; of metaphors casting abstract activities in terms of mundane activities, 63; of metaphors casting complex situations in terms of fights, 63; of metaphors casting moods in terms of heights, 63; of metaphors not belonging to a systematic family, 63; of metaphors used by metaphor-bashing philosophers, 22; of me-too’s featuring subtle conceptual slippages, 146; of me-too’s triggered by a compliment to a spouse, 147–148; of mistaken categorizations, 527; of monosyllabic lexical blends, 267; of morality-as-cleanliness stock phrases, 289; of “much” phrases, 67, 69, 70; of a multitasker’s activities, 403; of mundane, unseen analogies, 23; of negative numerical quantities in everyday life, 441; of 9/11’s, 297–298; of noncountability of members of the simplest of categories, 61; of non-lexicalized categories, 139; of non-subjective analogies, 525; of notions implicit in the category plate, 519; of objective categorizations, 522; of obsessions engendering analogies, 301–302; of office-like categories, 74; of old words with new technological meanings, 396, 398; of once bitten twice shy situations, 103; of opaque French idioms, 97; of operation–result “equations” in daily life, 408; of other cultures’ proverbs for “Once bitten, twice shy”, 105; of pairs of contradictory proverbs, 101; of parameters affecting one’s likelihood to jump to a conclusion via analogy, 307, 309; of Parises of the United States, 16; of parts of an airport, 52; of pasta types, 243–244; of phrases describing space in terms of time, 63; of phrases describing time in terms of space, 63; of phrases modified by “quote unquote”, 64–65; of physics phenomena belonging to electromagnetism, 467; of physics phenomena belonging to mechanics, 466, 494; of pieces of knowledge needed to understand a contemporary phrase, 128; of Platonic animal categories, 56; of plausible contributors to a lexical blend, 266; of pointless analogies, 283; of polysyllabic lexical blends, 267; of popes, 219; of possible meanings of Nick’s Nubian me-too, 151; of potential distinctions between categorization and analogy-making, 503; of potential members of the category bird, 59, 60; of proverbs about the sum of many little things, 109; of proverbs applicable to rationalization situations, 117–118; of proverbs poised halfway between categorization and analogy-making, 100; of pseudo-proverbs, 105; of questionable members of familiar categories, 528; of random thoughts in an airport, 33; of readymade sentences, 98; of reasons division always makes things smaller, 417; of reasons division word-problems cannot give a larger answer, 418; of reasons underlying Greece’s position in the Falkland Islands War, 332; of recent words coined from productive suffixes, 129; of reliable run-of-the-mill analogies, 529; of rival categorizations of a bag-toting woman, 127; of running-race metaphors, 289; of salient entities typically used in caricature analogies, 320; of sample languages that Google Translate offers, 377; of schoolday analogies, 17; of sci-fi fantasies rooted in familiar phenomena, 314; of sentences with the word “normalement”, 82; of similarities underpinning the Carol/Isabel analogy, 313; of single-word categories, 85; of situations defined by proverbs, 174; of sour grapes situations, 113, 114; of spicy one-line analogies, 136; of stock phrases shooting words into the speech chain, 25; of striking events liable to evoke memories, 158; of subcategories of dog, 240; of subjective categorizations, 523; of superordinate categories of dog, 240, 520; of symbols of sadness after a death, 300; of tail wagging the dog situations, 120-121; of thank-you’s of various flavors, 46; of things analogous to asparagus tips, 19; of trivial side show situations, 162–163; of types of analogies used in word- and phrase-retrieval in one’s native language, 376; of types of sandwiches, 214–216; of types of shadows, 204–209; of types of waves, 209–214; of unfamiliar concepts, 390; of unintended name slippages, 224; of universally important concepts, 80; of unstriking events not liable to evoke memories, 158; of unusual professions, 242; of usages of “much”, 70; of varieties of mess, 127; of verb-like phenomena presumably lacking mass, 475; of verbs used as category names, 66; of virtual actions frame-blended with real-world actions, 405–407; of virtual actions we perform daily, 395; of virtual objects we use daily, 395; of ways of missing the gist of a situation, 125; of wildly different-looking animals, 516; of words that start with “multi”, 413; of words with different connotations in English and Chinese, 368; of zeugmas, 6 literal encoding as inadequate, 174–175; see also abstraction literal-mindedness in the Copycat domain, 348–349, 351, 355, 357–360, 363–364 Locke, John, 22 logic: influence on psychologists’ theories of categories, 55, 436; the small role of, in thinking, 258, 288, 392; versus psychologic, 410; see also analogic versus logic loot carried off by thief, 472 “lovely spots” on city streets, as influencing perception, 296–297 lowercasing of categories, 34–35, 44 “lustre”, as possible French analogue to English word “score” in Gettysburg-address translation challenge, 372 —M— Macbeth effect, 289–290 Mach, Ernst, 487 machine translation, see translation Madonna, 223; syllogistic proof of mortality of, 193 magical angel stung by randomly buzzing interplanetary bumblebee, 493 magnet in motion, giving rise to electric field, 493 making distinctions and seeing commonalities, 189, 198 Malevich, Kazimir, 296 “man”, ambiguity of, 193–195 Mandarin, see Chinese Mandelbrot, Benoît, 444 manipulations, routine, in math, 449–450 manipulators versus manipulatees, 382–383 Marceau, Marcel, 322 marking, 186–187, 193–204, 217–218; applied to proper nouns, 227; and category extension, 254; helping to reveal a concept’s essence, 255; list of examples of, 195; in mathematics, 228–232, 419; in metaphor understanding, 229–232; origin of the term, 218 marriage, concept of, in constant evolution, 53 Martin, Mr.: as dog fancier, 238–239; as multi-categorizer, 189–190, 197, 248, 435 Mary, mother of Jesus, 38 Maslow, Abraham, 301 mass: barrier between two varieties of, 476, 478, 484; belonging to immaterial phenomena, 475; conservation of, 472, 475; interconvertibility between two varieties of, 480, 484; loss of, as result of radiation, 470–471, 475; normal versus strange, 476–485; poofing out of existence, 472, 475, 477–479, 481, 482, 484; possessed by energy, 471–478; possessed by heat, 475, 476; possesses energy, 482; two types of, 476, 477–478, 485 mass/energy analogy in Einstein’s mind, 472, 479–481, 482, 484; running aground on a fatal snag, 480, 484 mathematical formulas, mistaken views of, 391–394 mathematicians: arguing over category membership, 392; reluctant to extend categories, 440–444; seeing analogies between analogies, 502; “sniffing” the crux of a problem, 451; thinking by analogy, 439–451; toolkit of, 450 mathematics: ambiguity in, 237; analogy with Monopoly, 450–451; causality in, 411; imagined as lacking blurriness, 233, 392, 439; intuition in, 451; marking in, 228–232; naïve analogies in, 407–434, 439; rooted in everyday experiences, 393, 427; routine situations in, crying out for specific routine techniques, 449–450; spectrum of subtlety of analogy-making in, 451; unusual categorizations of, 510; use of analogy in, 439–451 matter as imbued with energy, 481 Maxwell, James Clerk, 130, 212, 213, 275, 361, 453, 459, 502 Maxwell–Boltzmann distribution, 457, 458; see also ideal gas Maxwell’s equations of electromagnetism, 361, 410–411, 456, 459, 489; both confirmed and undermined by one and the same experiment by Hertz, 460 measurement: contrasted with sharing, 420, 422–426; of energy E in units of size hν, 459; as key concept in division, 420 “Mecca” pluralized to “mecca”, 220 meccas, list of, 220; of wind-surfing, 229 mechanical translation, see translation mechanics: defined, 466; generalized to all of physics, 466–467, 494–495, 499 medium: in producing shadows, 207–208; of waves, 210–214 “melting” of components in compound words, 87, 111 membership in categories: non-black-and-white nature of, 14, 55–57; transitory nature of, 225 memo to office assistant, ambiguity of many words in, 395 memories of the past, as allegedly shackling people, 313–315 memory retrieval: alleged uselessness of, 338, 341; allegedly triggered by irrelevant features, 341; salient features’ dominance in, 342; surface-level features’ alleged dominance in, 337–346; virtuosity in, 110; see also remindings menace, as typical example of a verb naming a category, 66–67 mental blocks, recipes for escape from, 248–249 mental bridges, 183–184; see also analogy-making mental lexicon, 137 mental simulation in math word-problems, 421–425, 427–429 mental spaces, 365; see also Fauconnier mess, as example of a highly protean concept, 5, 127, 510 meta-analogies: in doing theoretical physics, 212; in evolution of wave concept, 211–212; in ordinary conversation, 27 metallurgists appreciating the blurriness of the category metal, 60 metaphorical versus literal meanings, 37–38 “metaphorical” usages: not always metaphorical, 229–230; three types of, 230–232 metaphors: conventional, 229–232; creativity in many, 510; embodiment and, 286–289; families of, 63; as flapdoodle, 22; Glucksberg–Keysar theory of, 228–229; going dead over time, 64; list of common words used as, 62; list of sentences using stock, 232; mixture of abstract and concrete in, 286–290; “mobile army of” (Nietzsche), 21; process of understanding of, 228–232; scorned by Hobbes, 22; used to criticize metaphors, 22 “me too”, unclear halo implicit in the phrase, 145, 150–153 me-too analogies, 143–153; in Copycat’s microdomain, 346–358; marginal members of the category, 147; phrases that often are giveaways of, 143, 152, 507; the ubiquity of, 507–508 métro in Paris, 215; American transculturation of, 377–379 microworlds, 305 “Mighty oaks from little acorns grow”, 109 military-budget arguments mobilizing flow of ideas, 26 military versus non-military analogies in times of war, 333–335 milk carton too heavy for bag, 133 Millikan, Robert, 461 Minkowski, Hermann, 453, 498–499 mistaken-identity scenes, 291–292 “mobile army of metaphors” (Nietzsche), 21, 509 Molière, 186, 248 Mommy as core of concept mother, 34–37 Mona Lisa with mustache, 351 monolithicness of categories, illusory, 3–5, 9–13, 71, 81–83, 241 Monopoly, 11, 450–451 Moon, analogical extension of, leading to the concept moon, 43–45, 64, 147, 210, 217 moonlets in Saturn’s rings, waves in medium of, 21 “morsel of shame”, 140 Moser, David, 89, 150–151, 259, 291–292 mosquito: in nudist colony, 320; perspective on Albert Einstein of, 163, 164–165 mother: abstract extensions of the category, 37–38, 53; development of the category, 34–38, 48, 53; marginal cases of, 37–38; as opposed to the concept mommy, 36 motion, children’s naïve view of, 294–295 Mount Analogy: scaling hardest slopes with and without pitons, 131; trekking on, 126, 131–132 mouse, as tangible gateway to intangible world, 252–253, 509 mouse/limb analogy, 252–253 Mozart, Wolfgang Amadeus, 223; of mushrooms, the, 222, 229 “much”: syntactic slots for, as a category, 68–70 much-situations: nature of, 67–68; role of expectations in, 68 multi-categorizability: of objects, 59, 189–192; of situations, 117–118, 188, 248 multiplication: apparent asymmetry of, 413, 415, 428; commutativity of, 413–416; generalized to abstract objects, 446–447; mental simulation used to solve, 424–425, 427–429; naïve analogies for, 411–416; as necessarily making larger, 413–414, 416; as repeated addition, 412–416, 427–429; tables, patterns in, 446–447 multitasking: concept borrowed without awareness from computer world, 402–403; passage of the concept into the everyday world, 404 Munich conference, 332, 334; pluralization of, 335 Murphy, Gregory, 60, 436 Mussolini, Benito, of mulligatawny, the, 222, 360 music keyboards, electronic, and musicianship, 131 musical instruments and zeugmas, 11–12 mystical characterization of genius, 501 —N— naïve analogies, 31–32; as bases for effective interfaces, 400; coexisting with other views, 389, 409; concerning analogy-making, 451; concerning categorization, 435–436; concerning cicadas, 388; concerning disk ejection, 401; concerning division, 416–421, 425–426; concerning email addresses, 385–387; concerning the equals sign, 407–411; concerning icons, 402; concerning motion, 294–295; concerning multiplication, 411–416; concerning shaving, 385–387; concerning size changes, 295; concerning titmice, 385–387; concerning virtual desktops, 401; deep entrenchedness of, 394, 409; defined, 386; education and, 389–394, 409, 411–434; like groomed-slope skiers, 389; linking cleanliness to morality, 289–290; made by analogy experts, 436–437; in mathematics, 407–434, 439; in mathematicians’ minds, 439, 441; misleading nature of, 400–401; not eliminated by scientific training, 389, 394; permeating today’s computer technology, 400; rooted in everyday experience, 386, 389, 391, 393–394; stemming from the computer world, 402–407; unconscious character of, 386, 389, 512; underlying jargon-creation, 399–400; utility of, 389 naïve equations, 407–411 names: conflated with items they name, 227; retrieved by analogy, 224–225 Napoleon: of fossil bones, the, 222; frame-blended with emperor penguins, 380 nature, cut at the joints by categories, 14, 77, 522–523 N-dimensional spaces, 443–444 nebula, as image for a language’s filling of a conceptul space, 119 necessary and sufficient criteria for category membership, 55, 436 negative numbers: fear of, 439–442; square roots of, 442–443 Neruda, Pablo, 522 nested radicals in polynomial solution-formulas, 445 Newman, Paul, 318 Newton, Isaac, 130, 210, 443, 471, 490, 491, 500; law of gravitation of, 389, 489; second law of motion of, 410, 491; of terrorism, the, 222 New York subway, as translation of Paris métro, 378 Nick’s me-too quip to the Nubian taxi driver, 151–152 Nixon, Richard: of superheroes, the, 222; yearining to be known as “RMN”, 90 Nietzsche, Friedrich, 21, 509 9/11, see September 11th Nobel Prize: citation for Einstein, extreme caution of, 461–462; for creative extensions of categories, 464–465 non-Euclidean geometry, 16, 498–499 non-lexicalized concepts, 30, 137, 139–140, 176–180 normalement, as monolithic concept in French, 82 normal mass/potential energy analogy, 480–481 normal mass/strange mass membrane, broken, 480–481, 482, 484 Norman, Donald, 259, 400 norms as directing word choices, 73 novices: inability to spot depth, 341; versus experts, 236–246, 255, 392–394 Nubians harmed by dam, me-too analogy centered on, 151–152 number : blurriness of the category, 392; relentless conceptual broadening of, 439–443, 447–448 numerical comparisons as analogies, 153–154, 281–282, 285, 331 —O— oars replaced by javelins, 317, 322 object recognition, mediated by analogy, 19, 184 objectivity: of analogies, 522–526; of categorization, 522–526 obsessions engendering analogies galore, 258, 299–305, 524 October 11th crash, 31, 297 “Ode to Constraints” (James Falen), 315 “office” versus “study”, 47–49, 74, 76 office visit as an example of a schema, 336–337 “official” boundaries of categories, 64–65 old town, as metaphor for a category’s core, 61–62, 65 Once bitten, twice shy: as an abstract category, 100, 103–106, 516; as a proverb incarnated in various languages, 105; as analogical pressure in column-translation dilemma, 306–307 “one”, as the name of a category, 75 one-dollar bill, as minimal banknote, 280 one-line analogies, list of, 135 one-member categories, see single-member categories “one smart dude”, as indicative of category of speaker, 75 opacity: of acronyms, 91–93; of compound words, 86–87; of idiomatic phrases, 97–98 operation–result naïve analogy for equations, 407–411; see also cause–effect Oppenheimer, Frank, 275 Oppenheimer, Robert, 275, 507 opposite meaning: produced through biplans, 268; produced through conceptual-proximity slippage errors, 276–277 Orwell, George, 57 —P— “pacifier”, semantic components of, unheard by toddler, 86 Pac-Man, obsession with, 303–305 pantheons, 219–220 paradoxes stemming from the alleged dominance of the superficial in retrieval, 341–344 parallels between parallels, Maxwell’s love of, 502 Paris: being Paris, 522; genericity of, to French people, 378; growth of, over centuries, 61; métro stations in, 215, 377–379; as tourist mecca versus book-writing locale, 163; of the United States, 16, 378; as venue of Marie Antoinette’s dizzy remark, 358 parking places in San Francisco: beauty of, 296–297; surprising availability of, 327–328 particle–antiparticle annihilation, 482 parts hidden inside wholes, 86–93 Pascal, Blaise, 101, 102 past, as key to understanding the present, 20, 23 pasta, expertise in, 243–244 Pasteur, Louis, 300 patent clerk, Third Class, 457, 460, 463, 470 “pathological” functions, 392 “Patsy is a pig”, see metaphors, “pig” pattern, as monolithic concept in English for which French has no single word, 81–82 patterns in discourse space, 69–76 patterns in multiplication tables of groups, 446–447 “peaks” of concepts poking out above clouds, 50, 52 Pearl Harbor, as category, 298 pedaling in sauerkraut, 248 pedestal, shared, as conceptual skeleton of two different word problems, 433–434 peel, semantic halo surrounding, 126, 270–271 people, analogically conflated, 181, 224–225, 275 perception: context-biased nature of, 299; dependency on concepts, 171; without concepts, 172, 315 perception of grammatical situations, 69–70 permutations, successive, as giving rise to groups, 446 personal celebrities, 222–223 Peter-Defare, Evelyne, 259 Peter miswriting the year every January, 148–150, 174 Phædrus, 112 Phædrus, 522–523 Phelps, Michael, 154–155, 367 philosophy of life, courtesy of Pac-Man, 303–305 phonetic proximity, role of in speech errors, 265 “phonon” as name for sound quantum, 459 photoelectric effect: behavior predicted by Einstein, 460–463; discovered by Hertz, 460; Einstein’s predictions confirmed by Millikan, 461; merely an afterthought in Einstein’s 1905 light-quantum paper, 460, 462 “photon” as name for light quantum, 459, 461, 462, 482 phrase choice constrained by sentence choice, 26 phrases: blended together, 259–265; retrieved by analogy, 93–98 physical world, understood via naïve analogies to computer world, 402–407 physicists: perception of equations by, 410–411; stereotype of, 451–452 physics: naïve analogies in, 410–411; seen as deductive, axiomatic discipline, 452; seen as generalization of mechanics, 467–468 physics problems, as perceived by novices versus experts, 342 physics thinking/political thinking analogy, 337 π, 302, 409, 410, 413, 444, 498 pianist: striking one wrong key, 270–271; striking two keys at once, 263, 266 “pig”, metaphorical use of, 228–232 pinball-machine obsession, 301 ping-pong, thanks to analogies, in discovery process, 500 pinpointing of essence, see essence-spotting Pisa: Galileo’s use of the tower of, 492, 493; prior to its famous tower, 319, 468; with tower not yet leaning, 472: with tower starting to lean, 482 piton-placing as metaphor for concept creation, 131 pizza consumption, as generic bland event that does not trigger remindings of specific events, 158–159 Planck, Max, 456–458, 460–461; disdain for Einstein’s light quanta, 460, 461, 463; likened to thirsty horse, 457; pardoning Einstein’s sins, 461; skeptical of existence of atoms, 460 Planck’s constant h, 456, 459 planet, as category requiring long deliberation to decide about membership in, 60, 512, 514, 528 plans, blending of, see biplans plastic card as a key, 254 plate, as category lacking relationships among non-parts, 518–519 plate-throwing woman, frame blend by, 367 Plato: of freemasonry, the, 222; objectivist vision of, 190, 522–523; warning of analogy’s slipperiness, 21 Platonic concepts, hopefully precise laws of, 56, 58 “play”, zeugmatically exploited, 10–12 pluralization: of Bible, 220; of famous people, 221–222, 254, 297; of friends or relatives, revealed by speech errors, 224; of friends, via strong resemblance, 224; of Hitler, 335; of Jeff, 223; of Mecca, 220; of Mommy, 34–35; of Moon, 44; of Munich, 335; of Pantheon, 219–220; of Pope, 219; of September 11th, 297–298; of signature-botching, 149; of a specific wine, 244 plurals of compound nouns, 87 Pluto, debate over its status as planet, 60, 512, 514, 528 poem learned by rote as member of category boat on tracks, 522 poems in the text: “Arizona Ants” (Kellie Gutman), 160, 381; “The Fox and the Grapes” (Benserade), 112; “The Fox and the Grapes” (La Fontaine), 112; “The Gardener’s Daughter” (Tennyson), 397; “Karnak Caps”, (Kellie Gutman), 160, 381; “La cigale et la fourmi” (La Fontaine), 388; “Ode to Constraints” (James Falen), 315; “Psalm XXX” (Milton), 397; “There Is No Word” (Tony Hoagland), 133 Poincaré, Henri, 132; on flesh of geese and of dogs, 132; letter of reference for Einstein by, 501; on mathematical thinking, 439–440, 509; sudden flash of inspiration of, 16 pointless analogies, see analogies, purposeless Poirot-Delpech, Bertrand, 373 political analogies, 17, 331–337 Polya, George, 507 polynomials: over finite fields, 447–448; imaginary numbers in, 448; search for general solution formula for, 445 pool table/ideal gas explanatory analogy, 457 Pope: of atheism, 219; pluralization of, 219; as salient entity used in caricature analogies, 320; of search engines, 220 positron (= anti-electron), 482 potential analogies, see semantic halos potential energy, 479–481 pressures: to categorize in real time, 258, 261; in creative translation, 371, 380–382; in Einstein’s mind, 477, 480–481, 485; guiding caricature analogies, 323; inducing fluid conceptual slippages in Copycat domain, 350–351, 352, 354–357; to make equations reflect cause and effect, 407–411; pushing for creative analogies, 300–301, 355–356, 458, 477, 480–481; see also cognitive dissonance prime numbers: generalized to “prime groups”, 449; generalized to “prime knots”, 449; generalized to primes inside rings, 448 primitive needs as primeval forces, 314 “prison”, metaphorical use of, 228–229 prison of the known, Krishnamurti’s putative 313–315 privileged category of each entity, 190, 435 probabilities, as hinted by strengths of analogies, 308 problem-solving: led astray by miscategorization, 293–295; mistaken for the raison d’être of analogy-making, 283, 285 Procrustes, bed of, 144–145, 160 productive suffixes “-holic”, “-thon”, and “-ism”, 129 professions, hierarchical structuring of, 242–243 proper nouns, pluralization of, 217–223 proportional analogies, 15–16; as gleaming jewels, 16; unhelpful in devising caricature analogies, 323–324; as unnatural view of most analogies, 144–145 proportionality/analogy proportional analogy, 15 proportionality to mass: of fictitious forces, 488; of gravitational forces, 489–491 prototype theory versus exemplar theory of concepts, 57 proverbs: families of, 109; as filters through which to understand situations, 101, 102; as names of categories, 100–102; non-opacity of, 106; objective reality of instances of, 110, 111, 132–133; overly general interpretations of, 107; recognized in situations, 174, 188; retrieval of, 104–105, 110; scope of, 106–109; surface versus essence of, 106–109; use of, as an act of analogy-making, 100; use of, as an act of categorization, 100 pseudo-proverbs, 105, 106 psychic trauma as a notion foreseen in the proverb “Once bitten, twice shy”, 104 psychological pressures leading one to map oneself analogically onto others, 153, 154–155 “psychology does not recapitulate etymology”, 86 public categories, 100 “pull no stops unturned”, as quintessential lexical blend, 262–264 pumpkins, pastries, plows, and pigs, 66 puns under attack, caricature analogy of, 319 “pure” versus “uncontaminated” analogies, 363–364, 366–367 Pushkin, Alexander Sergeevich, 130, 132; constraints in poetry of, 315; of feminism, the, 222 putting finger on a situation’s essence, see essence-spotting Pythagoras, as a category, 221 —Q— quadratic equation: broken up into six cases, 441; formula(s) for, 438, 441 quadrilaterals, classification of, 233–238, 255 quality control, as explanatory analogy, 329 quantum of energy: of electromagnetic wave, 459; of heat, 461; of sound, 461; of vibrating atom, 456–457, 461 quartic equation: group of symmetries of its solutions, 446; strange formula for, 445 “quatre-vingts” as translation of “four score”, 370–371 quintic equation: search for formula for solutions of, 445–446; unsolvability via radicals proven for, 446 quotation marks: as a convention of this book for words, 34, 110; for honorary category members, 44, 64–65; second-order, 65 “quote unquote”, as way of indicating metaphorical usage, 64–65 quotient groups, 448–449 quotient skyscrapers, 448 —R— Raban, Jonathan, 284 random murder as conceptual skeleton, 248 random resemblances constantly noticed, 284 randomly buzzing interplanetary bumblebee, see magical angel rapid right-on retrieval: as the core of cognition, 127; as the essence of intelligence, 125–126; as needed for survival, 79, 83, 505–506; see also essence-spotting rationalization and sour grapes, 115–118 read ⇒ write conceptual slippage, 276–277 reading, as triggering ideas in a mind, 376–377 ready-made sentences as categories, 98–99 Reagan, Nancy, 358 reality of members of abstract categories, 110, 111, 132–133 reasoning, as opposed to analogy-making, 333; see also logic, analogic versus logic recategorization of situations, 73, 249–252, 327–328 reclothing a stripped-down essence, 153 Recorde, Robert, 408 reduction ad absurdum technique in mathematics, 450 redwood trees, trip to, 310–312 refinement of categories, as reaching a limit, 83 relationships among parts: as crucial for analogy-making, 517–518; as crucial for categorization, 518–519 relativity, Galilean, principle of, 466–468, 485, 486, 492 relativity, general: analogies at root of, 491–495, 499; attempts at, 490–491; experimental confirmation of, 496; goals of, 486–488; rotating disk in, 497–498 relativity, special, 361; analogy at root of, 467–468 remindings: as crucial for survival, 172–173; as a deep mystery of cognition, 159–166, 354; as due to analogousness, 18, 30, 336; idiosyncratic nature of, 525–526; induced by traumatic experience, 225; mediated by faces, 181–184; mediated by identical encodings, 173; mediated by many diverse cues, 171; opacity of mechanisms of, 511; revealing the existence of unsuspected categories, 168; seeming not to need explanation, 18; triggered by simple visual analogies, 169–170 repeated addition: as crux of multiplication, 412–416; as way of solving multiplication problems, 427–429 reporter #1/reporter #2 romantic analogy, 305–306, 308 retrieval of memories, see remindings, triggering reversal: by Einstein, 474, 482–483; as potential source of humor, 280; role of, in creativity, 356–357, 363–364, 371 rhyme, preservation of, in poetry translation, 381 rich and poor zones of a language in conceptual space, 82–83 Richard, Jean-François, 294–295 Riemann, Bernhard, 498 “right” versus “wrong”: in analogy-making, 16; in Copycat domain, 350–351, 352; see also esthetics Ringfinger, Renate, 464 rings, as homes of new types of numbers, 448 ripples, see waves Rips, Lance, 390 rival analogies: in real-time competition, 260–278; in wartime decision-making, 333 rock-climbing as metaphor for creative thinking, 131 rock music, category in the mind of a classical-music lover, 241 role reversal in Grand Canyon episode, 163, 165 Roosevelt, Franklin Delano, 90, 275 roots of polynomials, see solutions of polynomials rope, speaking of, in the house of the hanged, 104, 311 Rosch, Eleanor, 55, 345, 436 Rossi, Mario, 259 rotating disk/non-Euclidean geometry analogy, 498 rotations of a cube, as number-like entities, 446–447; see also groups royalty statement triggering analogies, 153–154 Ruffini, Paolo, 446 rule of thumb separating analogy-making and categorization, 515 Rumelhart, David, 259 Russian language, 9–10, 12, 368; Anna’s dream in, 504; “but” in, 74 Ruth, Babe: of bank robbers, the, 222; 1927 Yankees minus, 468 Ruth, Dan’s image of, contaminated by Jeanine, 225 Rutherford, Ernest, 143 —S— sabbatical year, zooming in on details of, 50 Sagan, Françoise, obituary of, as translation challenge, 373–377 salience: of any feature as subjective, 363–364; of deep features to experts, 342–344 salient features dominate in memory retrieval, 342 salsa: debugging of technique in, 403–404; the pope of, 219 salt/sugar confusion as categorization error, 102, 527 Sander, Emmanuel: as error collector, 259; explaining humps and bottles to his son, 198–200; falling momentarily for categories = boxes, 436; making analogy between co-author’s two blue station wagons, 283; as one-time Pac-Maniac, 303–305; smiling with joy at finally finishing book :), 575; taking coffee break, 185, 317; transculturated to San Francisco, 327–328 Sander, Mica, 40, 198–200, 295 Sander, Talia, 17, 39, 40, 43 Sander, Tom, 40, 126, 233–234, 236 Sandwich, Earl of, the fourth, 214 sandwiches: “A–B–A” form of, 215–216; abstraction of, 214–216; of appointments, 216; blurry boundaries of category, 214–216; bread role in, 214–216; edible, 214, 216; horizontality of, 215; meat role in, 214–216; in Paris métro, 215; in physics, 215–216; of rhymes; 215; sexual, 215; symmetry, as unclear criterion in, 215–216; transistors as, 215; walking, 214–215 sandwichology, burning questions of, 215–216 San Francisco, parking in, 296–297, 327–328 Santa Clara Valley, metamorphosis of, 397 Sapir–Whorf effect, 123–124; cultural version of, 128–131 Saturn’s rings, waves in the medium of, 213 savanna, 71, 364–366 Schank, Roger, 104, 173 schemas: as another name for categories, 336; office visit as an example of, 336–337; versus concrete concepts, 336–337 Schrödinger, Erwin, 453 Schweitzer, Albert, face of, 183–184 science-fiction story as core of a category, 524–526 scientific discoveries: boldness of analogies in, 360–361; mediated by seeing two phenomena as bagels from the same batch, 310 Scott/Thor facial resemblance as an analogy, 181–182 search engines, limited to surface, 115 search, virtual, frame-blended with physical search, 402, 405 secret agent in tunnel category, 167–168 self-monitoring by speakers, 72–73 Selvinsky, Il’ya L’vovich, poem by, 9–10 semantic approximations, 39–43, 270–278 semantic halos: errors caused by, 270–278; as sources of latent analogies, 271, 273 semantic memory, 137 semantic space/nebula analogy, 119–120 semantic space, zones in, 10, 78–81, 83–84, 118–124, 132; see also conceptual spaces senses, physiological, and analogy-making, 286–288 sentence choice constrained by idea choice, 26 sentences: blended together, 268–269; ready-made, 98–99 September 11th: as category, 297; imposing itself on perceptions of events, 31, 297–298; pluralization of, 297–298 sexist default assumption, 293 sexist language and marking, 193–195 shadow: due to absence of light, 204–206; due to absence of mysterious particles, 208; due to absence of rain, 205, 207; due to absence of snow, 205–206; due to absence of vehicles, 207; due to absence of young males, 208; gradual abstraction of, 204–209; in late afternoon, 205; of Nazism, 208 Shakespeare, William, 130, 132; of advertising, the, 222 shallow depth, 346 shallower and deeper aspects of concepts, 203–204 shallow features, experts’ blindness to, 343–344 sharing: contrasted with measuring, 420–426; as key concept in division, 419–426; marked sense of, 419; as necessarily reducing, 419 shells in a conceptual space, 81 shoes: of Albert Einstein, 455; left versus right, 427 showers, used by analogy, 23, 507, 509 sibling, concept of in various languages, 77 silver platters, analogies handed to the reader on, 160, 170 Simmons, Curt, 325 simplification, as key drive in mathematics and physics, 440 simulation, see mental simulation single-member categories as no different from multiple-member categories, 39 “sitting right there”, 140–141 situations: constant real-time encoded of, 161; doing the thinking in math problems, 432; evoking categories, 45–47, 450; lacking clear boundaries, 33, 161; multi-categorizability of, 188; possessing both superficial and deep aspects, 342–344, 515; see also analogy-making, remindings sixty, pointless analogy involving, 281–282, 285 size changes, adults’ naïve view of, 295 size, role of, in encoding of situations, 163 skunk caused by stench, thanks to Maxmell’s equations, 411 slippages, conceptual: between opposite concepts, 276–277, 356–357; in caricature analogies, 321–326; cascade of, 357; due to conceptual proximity, 270–278; engendering conceptual broadening, 150; in level of abstraction, 186; in me-too analogies, 144, 146–148, 151; riding on coattails of other slippages, 276, 357; role of, in creativity, 186–187, 249–256; triggered by esthetic pressures, 350–351, 352, 354, 357; unintended, from one person’s name to another’s, 224 Smith, Peter, see Peter miswriting year smoking causing impotence, 362 smurfs, limited vocabulary of, 108 snag, outflanking of, in Copycat domain, 356–357 “sniffing” the crux of a math problem, 450–451 Snoopy the cat, caricature analogy involving, 319 snow shadow, photo of, 206 snuoiqers versus iggfruders, 11 “so to speak”, to indicate honorary category members, 64–65 soccer played with a bowling ball, 318 Socrates, 16; of snails, the, 222 solutions of polynomials, formal symmetries of, 446–447 sound choice mediated by word choice, 25 sound particles/light particles analogy by Einstein, 461; see also light waves/sound waves analogy sound-quantum hypothesis, 461 sound waves, Doppler effect for, 469–471 sounds versus noises, 126 source–target paradigm in psychology experiments, 339–340 sour grapes situations: category of, 29–30, 113–118, 310; contrasted with silver lining situations, 117–118 space/space-time analogy, 498–499 Spalding, Thomas, 436 Spanish language, 369, 522 speaker/driver analogy, 73 special relativity, see relativity, special spectrum: blackbody, 455–459; defined, 455–456; of ideal gas, 457–459 speech errors: blatant when placed in frames, 261; collecting of, 261; no extra insights in analyzing one’s own, 264; rampant on Web, 261; revealed by hesitations, phonetic distortions, etc., 263, 269, 281; translation of, 379; see also errors, lexical blends spider, as occasional member of category insect, 58 spilling the beans as a category, 96–97 spinning universe, 487 Spitz, Mark, 154–155, 367 square roots of negative numbers: analogy to ordinary numbers, 442; fear of, 442–443 squares, as questionable rectangles, 234–238, 255 staircases, negotiated by analogy, 507, 509, 516 Stargell, Willie, 325–326, 383 statistical approach to machine translation, 372–374 staying on the surface versus going into depth, 344 stealing, conceptual halo around, 106–107 stereotypes: of analogy-making, 135–136, 392, 521, 529; of creativity in physics, 452; efficiency of, 466; as overhasty categorizations, 527–528; shallowness of, 346 Stevens, Wallace, 38 sticks for stirring coffee, absurdity of, 317, 321–322 strange mass: analogous to energy, 479; mutating from one form to another, 479; versus normal mass, 476–485 Streep, Meryl, of spitting, the, 222, 360 strings, alphabetic, 347 string/wire conceptual conflation, 277, 278 “study” versus “office”, 47–49 stupidity, not the same as ignoring most of the world, 426–427 subgroups, nesting patterns of, 447 subjectivity: of analogies, 522–526; of categorizations, 522–526 subscripts/exponents analogy, 169–170, 174 substitutions, Lagrange’s theory of, 446, 447 subtraction word-problems, various strategies for solving, 421–422, 425, 429–434 suburban sprawl likened to marginal or metaphorical uses of a word or phrase, 62, 65–66 Sue (fictional Tim’s fictional mother), 34, 37, 38 sunset, as seen by astronomy students, 389 “superficial”: meaning of, 340; pejorative versus neutral connotations of, 344 superficial features: guiding perception only in one’s domains of incompetence, 340; experts’ blindness to, 343–344; role played by, in memory retrieval, 171, 343; versus deep features, 340 “superfluid” Copycat analogy, 352 superimposing of instances creating more abstract concept, 23, 35, 334, 336–337, 521–522; see also schemas surfaces: bad reputation of, 344; as cues to depths, 345–346; as royal road to essences, 344–346 surfaces versus essences: proverbs about, 102; of certain proverbs, 102, 106–107 surface/depth correlation, 345–346 surface/depth distinction: merely a surface-level contrast, 344; nonexistent for novices, 341–344 surgeon riddle, 293 surgery, mathematical notion of, 426 survival: as dependent on rapid analogy-making, 506–507; as dependent on rapid categorization, 79, 505–506 SUV/search engine analogy, 402 swerves in discourse space, 72–73 swimming pool/black body explanatory analogy, see black body “swimming pool table” analogy, 455, 457–458; see also black body/ideal gas analogy sword of Damocles, as a category, 95–96 syllepses, see zeugmas syllogisms, 15–16, 193, 437 symbol-manipulation recipes, role of analogy-making in the evocation of, 451 symmetry: abstract forms of, 446–447; as an ideal kind of analogy, 357 synopsis of the book, 29–32 syntactic slots as categories, 68–70 —T— tags for photos, as analogues to encodings of experiences, 172 tail wagging the dog concept, 120–121 tango rote memorization as member of category boat on tracks, 521–522 Tartaglia, Niccolò, 438 taste, good versus bad, see good taste technical terms, originating in everyday world, 395–400 technology, understood through homey analogies, 394–400 technomorphism, 404–407 telephone-answering gaffe, 175 “temps”, distinct concepts associated with the French word, 78 Thagard, Paul, 330 Thank you!

Below are listed some concepts — just a minuscule subset of the concepts that our culture abounds in — the possession of which would seem to give us a substantial leg up on people from previous generations or centuries: Positive and negative feedback, vicious circle, self-fulfilling prophecy, famous for being famous, backlash, supply and demand, market forces, the subconscious, subliminal imagery, Freudian slip, (Edipus complex, defense mechanism, sour grapes, passive-aggressive behavior, peer pressure, racial profiling, ethnic stereotype, status symbol, zero-sum game, catch-22, gestalt, chemical bond, catalyst, photosynthesis, DNA, virus, genetic code, dominant and recessive genes, immune system, auto-immune disease, natural selection, food chain, endangered species, ecological niche, exponential growth, population explosion, contraception, noise pollution, toxic waste, crop rotation, cross-fertilization, cloning, chain reaction, chain store, chain letter, email, spam, phishing, six degrees of separation, Internet, Web-surfing, uploading and downloading, video game, viral video, virtual reality, chat room, cybersecurity, data mining, artificial intelligence, IQ, robotics, morphing, time reversal, slow motion, time-lapse photography, instant replay, zooming in and out, galaxy, black hole, atom, superconductivity, radioactivity, nuclear fission, antimatter, sound wave, wavelength, X-ray, ultrasound, magnetic-resonance imagery, laser, laser surgery, heart transplant, defibrillator, space station, weightlessness, bungee jumping, home run, switch hitter, slam-dunk, Hail Mary pass, sudden-death playoff, make an end run around someone, ultramarathon, pole dancing, speed dating, multitasking, brainstorming, namedropping, channel-surfing, soap opera, chick flick, remake, rerun, subtitles, sound bite, buzzword, musical chairs, telephone tag, the game of Telephone, upping the ante, playing chicken, bumper cars, SUVs, automatic transmission, oil change, radar trap, whiplash, backseat driver, oil spill, superglue, megachurch, placebo, politically correct language, slippery slope, pushing the envelope, stock-market crash, recycling, biodegradability, assembly line, black box, wind-chill factor, frequent-flyer miles, hub airport, fast food, soft drink, food court, VIP lounge, moving sidewalk, shuttle bus, cell-phone lot, genocide, propaganda, paparazzi, culture shock, hunger strike, generation gap, quality time, Murphy’s law, roller coaster, in-joke, outsource, downsize, upgrade, bell-shaped curve, fractal shape, breast implant, Barbie doll, trophy wife, surrogate mother, first lady, worst-case scenario, prenuptial agreement, gentrification, paradigm shift, affirmative action, gridlock, veganism, karaoke, power lunch, brown-bag lunch, blue-chip company, yellow journalism, purple prose, greenhouse effect, orange alert, red tape, white noise, gray matter, black list… Not only does our culture provide us with such potent concepts, it also encourages us to analogically extend them both playfully and seriously, which gives rise to a snowballing of the number of concepts.

The Sum of All Fears by Tom Clancy


accounting loophole / creative accounting, airport security, Benoit Mandelbrot, British Empire, colonial exploitation, complexity theory, cuban missile crisis, demand response, financial independence, index card, mandelbrot fractal, trade route, uranium enrichment

The KGB's Eighth Chief Directorate is tasked to communications intelligence and communications security. 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. It was an historical accident that his father happened to be a General in the KGB's Chief Border Guards Directorate, and that as a result the Committee for State Security had taken immediate note of his work.

"So do the Americans. Get that message off at once. Then, I want everything we have from THISTLE on my desk." Golovko hung up and looked at the major standing in front of his desk. "That mathematician who figured this all out - good God, I wish we'd had him five years ago!" "He spent ten years devising this theory on ordering chaos. If it's ever made public, he'll win the Planck Medal. He took the work of Mandelbrot at Harvard University in America and MacKenzie at Cambridge, and -" "I will take your word for it, Major. The last time you tried to explain this witchcraft to me I merely got a headache. How is the work going?" "We grow stronger every day. The only thing we cannot break is the new CIA system that's starting to come on line. It seems to use a new principle. We're working on it." President Fowler boarded the Marine VH-3 helicopter before the snow got too bad.